operations with fractions problem solving

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Fractions operations

Here you will learn about fractions operations, including how to add, subtract, multiply and divide with fractions.

Students will first learn about fractions operations as part of number and operations in fractions in elementary school. They will continue to build on this knowledge in the number system in 6th grade and 7th grade.

What are fractions operations?

Fractions operations are when you add, subtract, multiply or divide with fractions.

For example,

[FREE] Fraction Operations Worksheet (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of fraction operations. 10+ questions with answers covering a range of 4th to 6th grade fraction operations topics to identify areas of strength and support!

Adding and subtracting fractions

Adding and subtracting fractions means finding the sum or the difference of two or more fractions.

In order to do this, the fractions must have a common denominator (bottom number).

The numerator shows the number of parts out of the whole and the denominator shows how many equal parts the whole is divided into.

To add or subtract the numerators (top numbers) and keep the denominators the same.

The equation is taking \, \cfrac{1}{8} \, away from \, \cfrac{4}{8} \, .

Since the denominators are the same, the parts are the same size.

You subtract to see how many parts are left: 4-1 = 3.

There are 3 parts and the size is still eighths, so the denominator stays the same.

When fractions have unlike denominators, create equivalent fractions with common denominators to solve.

The parts are NOT the same size, since the denominators are different.

Use equivalent fractions to create a common denominator of 10.

Multiply the numerator and denominator of \, \cfrac{2}{5} \, by 2.

\cfrac{2 \, \times \, 2}{5 \, \times \, 2}=\cfrac{4}{10}

Add to find how many parts there are in all: 2 + 4 = 6.

There are 6 parts and the size is still tenths, so the denominator stays the same.

The sum could also be written as the equivalent fraction \, \cfrac{3}{5} \, .

Step-by-step guide: Adding fractions

Step-by-step guide: Subtracting fractions

Step-by-step guide: Adding and subtracting fractions

Multiplying and dividing fractions

Multiplying and dividing fractions means using multiplication and division to calculate with fractions. Fraction multiplication and division can be solved using models or an algorithm.

Using models:

In the model, \, \cfrac{2}{3} \, is yellow and \, \cfrac{1}{2} \, is blue.

The product is where the fractions overlap in green.

The model shows \, \cfrac{2}{3} \, of \, \cfrac{1}{2}, \, so \, \cfrac{1}{2} \times \cfrac{2}{3} = \cfrac{2}{6} \, .

Using the algorithm:

To multiply fractions , you multiply the numerators together, and multiply the denominators together:

\cfrac{1}{2} \, \times \, \cfrac{2}{3}=\cfrac{2}{6} \, .

You can also divide fractions with a model or an algorithm.

Think of this equation as how many \, \cfrac{1}{4} \, fit into \, \cfrac{1}{2} \, .

In the model, \, \cfrac{1}{2} \, is orange and \, \cfrac{1}{4} \, is yellow.

To divide into equal groups, use the equivalent fraction \, \cfrac{2}{4} \, .

The quotient is the final fraction formed when \, \cfrac{2}{4} \, is put into a group of \, \cfrac{1}{4} \, .

Two groups of \cfrac{1}{4} can be made, so \cfrac{1}{2} \div \cfrac{1}{4}=2.

KEEP the first fraction, FLIP the second fraction, CHANGE to multiplication.

\cfrac{1}{2} \div \cfrac{1}{4}

Keep the dividend (first fraction): \, \cfrac{1}{2}

Take the reciprocal of the divisor (flip the second fraction): \, \cfrac{1}{4} \rightarrow \cfrac{4}{1}

Change to multiplication: \, \cfrac{1}{2} \times \cfrac{4}{1}

Multiply the fractions: \, \cfrac{1}{2} \times \cfrac{4}{1}=\cfrac{4}{2} \, which simplifies to 2.

\cfrac{1}{2} \div \cfrac{1}{4}=2

Since \, \cfrac{1}{2} \, is larger than \, \cfrac{1}{4} \, , the answer makes sense.

A larger number divided by a smaller number, will have a quotient of greater than 1.

Notice that it is not necessary to create a common denominator to multiply and divide fractions when using the algorithm, like it is to add and subtract fractions.

Step-by-step guide: Multiplying fractions

Step-by-step guide: Dividing fractions

Step-by-step guide: Multiplying and dividing fractions

The algorithm for dividing fractions involves using the reciprocal .

When two numbers are multiplied by something other than 1, and have a product of 1, they are reciprocals.

This is also known as the multiplicative inverse.

The reciprocal of all numbers can be found by writing the number as a fraction and then flipping it so that the numerator becomes the denominator and the denominator becomes the numerator.

Step-by-step guide: Reciprocal

Step-by-step guide: Multiplicative inverse

Common Core State Standards

How does this relate to 4th grade math, 5th grade math, and 6th grade math?

• Grade 4 – Number and Operations – Fractions (4.NF.B.3a) Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
• Grade 4 – Number and Operations – Fractions (4.NF.B.3c) Add and subtract mixed numbers with like denominators, for example, by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
• Grade 4 – Number and Operations – Fractions (4.NF.B.4b) ​​Understand a multiple of \, \cfrac{a}{b} \, as a multiple of \, \cfrac{1}{b} \, , and use this understanding to multiply a fraction by a whole number.
• Grade 5 – Number and Operations – Fractions (5.NF.A.1) Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, \, \cfrac{2}{3} + \cfrac{5}{4} = \cfrac{8}{12} + \cfrac{15}{12} = \cfrac{23}{12} \, . \; ( In general, \, \cfrac{a}{b} + \cfrac{c}{d} = \cfrac{(ad \, + \, bc)}{bd} \, . )
• Grade 5 – Number and Operations – Fractions (5.NF.B.4b) Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
• Grade 6 – Number System (6.NS.A.1) Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

How to use fractions operations

There are a lot of ways to use fractions operations. For more specific step-by-step guides, check out the fraction pages linked in the “What are fractions operations?” section above or read through the examples below.

Fractions operations examples

Example 1: adding fractions with like denominators.

Solve \, \cfrac{5}{8}+\cfrac{7}{8} \, .

Add or subtract the numerators (top numbers).

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 5 + 7 = 12.

2 Write your answer as a fraction.

There are 12 parts, and the size is still eighths, so the denominator stays the same.

\cfrac{12}{8} \, is an improper fraction and converts to the mixed number 1 \, \cfrac{4}{8} \, .

\cfrac{5}{8}+\cfrac{7}{8}=1 \cfrac{4}{8}

You can also write this answer as the equivalent mixed number \, 1 \cfrac{1}{2} \, .

Example 2: subtracting fractions with unlike denominators

Solve \cfrac{6}{10}-\cfrac{1}{3} \, .

Create common denominators (bottom numbers).

Since \, \cfrac{6}{10} \, and \, \cfrac{1}{3} \, do not have like denominators, the parts are NOT the same size.

Multiply the numerator and denominator by the opposite denominator to create equivalent fractions with common denominators.

\cfrac{6 \, \times \, 3}{10 \, \times \, 3}=\cfrac{18}{30} \quad and \quad \cfrac{1 \, \times \, 10}{3 \, \times \, 10}=\cfrac{10}{30}

Now use the equivalent fractions to solve: \, \cfrac{18}{30}-\cfrac{10}{30} \, .

Since the denominators are the same, the parts are the same size. You subtract to see how many parts are left: 18-10 = 8.

Write your answer as a fraction.

There are 8 parts and the size is still thirtieths, so the denominator stays the same.

\cfrac{18}{30}-\cfrac{10}{30}=\cfrac{8}{30}

You can also write this answer as the equivalent fraction \, \cfrac{4}{15} \, .

Example 3: multiplying a mixed number by a fraction with the algorithm

Solve 1 \, \cfrac{11}{12} \times \cfrac{3}{4} \, .

Convert whole numbers and mixed numbers to improper fractions.

Convert the mixed number to an improper fraction.

Multiply the numerators together.

\cfrac{23}{12} \times \cfrac{3}{4}=\cfrac{69}{}

Multiply the denominators together.

\cfrac{23}{12} \times \cfrac{3}{4}=\cfrac{69}{48}

If possible, simplify or convert to a mixed number.

The numerator is greater than the denominator, so the improper fraction can be converted to a mixed number.

\cfrac{69}{48}=1 \, \cfrac{21}{48}

The product can be simplified. 21 and 48 have a common factor of 3.

\cfrac{21 \, \div \, 3}{48 \, \div \, 3}=\cfrac{7}{16}

So, \, \cfrac{23}{12} \times \cfrac{3}{4}=\cfrac{69}{48} \, or 1 \, \cfrac{7}{16} \, .

Example 4: dividing a fraction by a fraction

Divide the numbers \, \cfrac{1}{12} \div \cfrac{1}{4} \, .

Take the reciprocal (flip) of the divisor (second fraction).

\cfrac{1}{4} \, → \, \cfrac{4}{1}

Change the division sign to a multiplication sign.

\cfrac{1}{12} \, \times \, \cfrac{4}{1}

Multiply the fractions together.

\cfrac{1}{12} \, \times \, \cfrac{4}{1}=\cfrac{4}{12}

\cfrac{4}{12}=\cfrac{1}{3}

This can also be solved with a model.

You can think of this equation as how many \, \cfrac{1}{4} \, fit into \, \cfrac{1}{12} \, .

In the model, \, \cfrac{1}{12} \, is yellow and \, \cfrac{1}{4} \, is orange.

To divide into equal groups, the fractional pieces need to be the same size.

Use \, \cfrac{1}{12} \, and \, \cfrac{3}{12} \, to solve.

The quotient is the final fraction formed when \, \cfrac{1}{12} \, is put into groups of \, \cfrac{3}{12} \, .

One out of the three parts are filled, so \, \cfrac{1}{12} \div \cfrac{3}{12}=\cfrac{1}{3} \, .

Example 5: adding mixed numbers with unlike denominators

There are 2 \, \cfrac{1}{3} \, pounds of red apples and 4 \, \cfrac{1}{6} \, pounds of green apples.

How many pounds of apples are there in all?

Create an equation to model the problem.

2 \cfrac{1}{3}+4 \cfrac{1}{6}= \, ?

Add or subtract the whole numbers.

Since \, \cfrac{1}{3} \, and \, \cfrac{1}{6} \, do not have like denominators, the parts are NOT the same size.

Use equivalent fractions to create a common denominator.

A common denominator of 6 can be used.

Multiply the numerator and denominator of \, \cfrac{1}{3} \, by 2 to create an equivalent fraction.

\cfrac{1}{3}=\cfrac{1 \, \times \, 2}{3 \, \times \, 2}=\cfrac{2}{6} \quad and \quad \cfrac{1}{6}

Add or subtract the fractions.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 2 + 1 = 3.

There are 3 parts, and the size is still sixths, so the denominator stays the same.

Write your answer as a mixed number.

Add the whole number and fraction together.

You can also write this answer as the equivalent mixed number 6 \, \cfrac{1}{2} \, .

There are 6 \, \cfrac{1}{2} \, pounds of apples in all.

Example 6: word problem dividing with fractions

Each seed needs \, \cfrac{1}{5} \, cup of soil. How many seeds can be planted with 11 cups of soil?

11 \div \cfrac{1}{5}= \, ?

Change any mixed numbers to an improper fraction.

Change 11 to an improper fraction.

11=\cfrac{11}{1}

Take the reciprocal (flip) of the divisor (second fractions).

\cfrac{1}{5} \, → \, \cfrac{5}{1}

\cfrac{11}{1} \times \cfrac{5}{1}

\cfrac{11}{1} \times \cfrac{5}{1}=\cfrac{55}{1}

If possible, simplify or convert to a mixed number (mixed fraction).

\cfrac{55}{1}=55

55 seeds can be planted with 11 cups of soil.

Teaching tips for fractions operations

• Fraction work in lower grades emphasizes understanding through models, including area models and number lines. To support students in upper grades, always have digital or physical models available for students to use as they work with fractions operations.
• Fraction worksheets can be useful when students are developing understanding around basic operations with fractions. However, when students have successful strategies and can flexibly operate, make the practice more engaging by using math games or real world projects that allow students to use fractions in a variety of situations.
• Highlight patterns within and between the operations as students are learning and encourage them to look for patterns on their own. This will help students make sense of the algorithms used to operate with fractions and minimize conceptual errors.
• Let students find reciprocal numbers on their own by consistently asking questions such as, “What number multiplied by 7 will have a product of 1 ?” Each time this is discussed, write these equations on an anchor chart and students will begin to see a pattern over time. Although worksheets can serve a purpose and help with skill and test prep practice, having students discover and make sense of mathematical concepts is more meaningful for building long lasting understanding.

Easy mistakes to make

• Forgetting how to find the reciprocal of a whole number Whole numbers can be written as an improper fraction and then the numerator and denominator of the improper fraction can be flipped to find the reciprocal of the whole number. For example, 16 can be written as \, \cfrac{16}{1} \, and the reciprocal is \, \cfrac{1}{16} \, .

Practice fractions operations questions

1. Solve \, \cfrac{5}{9}+\cfrac{2}{9} \, .

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 5 + 2 = 7.

There are 7 parts and the size is still ninths, so the denominator stays the same.

\cfrac{5}{9}+\cfrac{2}{9}=\cfrac{7}{9}

2. Solve \, 2 \, \cfrac{3}{10}-1 \, \cfrac{4}{5} \, .

The equation is taking \, 1 \cfrac{4}{5} \, away from \, 2 \cfrac{3}{10} \, .

Start with the fractions. Since \, \cfrac{3}{10} \, and \, \cfrac{4}{5} \, do not have like denominators, the parts are NOT the same size.

Use equivalent fractions to create a common denominator. Both denominators are multiples of 10.

\cfrac{3}{10} \quad and \quad \cfrac{4 \, \times \, 2}{5 \, \times \, 2}=\cfrac{8}{10}

Now use the equivalent fraction to solve: 2 \, \cfrac{3}{10}-1 \, \cfrac{8}{10}

However, there are not enough parts to take 8 away from 3.

You can break one of the wholes into \cfrac{10}{10} \, …

Now you can solve 1 \, \cfrac{13}{10}-1 \, \cfrac{8}{10}.

You subtract to see how many parts are left: 13-8 = 5.

There are 5 parts and the size is still tenths, so the denominator stays the same.

\cfrac{13}{10}-\cfrac{8}{10}=\cfrac{5}{10}

Subtract the whole numbers.

1 \, \cfrac{13}{10}-1 \, \cfrac{8}{10}=\cfrac{5}{10}

You can also write this answer as the equivalent fraction \, \cfrac{1}{2} \, .

3. Solve \, \cfrac{1}{4} \times \cfrac{2}{5} \, .

To solve using a model, draw a rectangle. Divide one side into fourths.

Divide the other side into fifths.

Shade in \, \cfrac{1}{4} \, with yellow and \, \cfrac{2}{5} with blue.

The model shows \, \cfrac{2}{5} \, of \, \cfrac{1}{4} \, , so \, \cfrac{1}{4} \times \cfrac{2}{5}=\cfrac{2}{20} \, , because there are 2 green squares and the whole has 20 squares in total.

The product can be simplified. Both 2 and 20 have a factor of 2, so they can be divided by 2 :

\, \cfrac{2 \, \div \, 2}{20 \, \div \, 2}=\cfrac{1}{10} \, .

So, \, \cfrac{1}{4} \times \cfrac{2}{5}=\cfrac{2}{20} \; or \; \cfrac{1}{10}

4. Solve \, 2 \, \cfrac{1}{6} \div 1 \, \cfrac{2}{3} \, . Write the quotient in lowest terms.

Change the mixed numbers to improper fractions:

Keep the dividend (first fraction): \, \cfrac{13}{6}

Take the reciprocal of the divisor (flip the second fraction): \, \cfrac{5}{3} → \cfrac{3}{5}

Change to multiplication: \, \cfrac{13}{6} \times \cfrac{3}{5}

Multiply the fractions: \, \cfrac{13}{6} \times \cfrac{3}{5}=\cfrac{39}{30}

Change back into a mixed number: \, \cfrac{39}{30}=1 \, \cfrac{9}{30}

Simplify: \, \cfrac{9 \, \div \, 3}{30 \, \div \, 3}=\cfrac{3}{10} \, , so the answer in lowest terms is \, 1 \, \cfrac{3}{10} \, .

5. Rashad is cutting a 12 \, ft rope into smaller \, \cfrac{2}{3} \, ft pieces. How many smaller pieces of rope will he have?

8 smaller pieces of rope

12 smaller pieces of rope

18 smaller pieces of rope

\cfrac{24}{3} smaller pieces of rope

Use the equation \, 12 \div \cfrac{2}{3}= \, ?

Draw 12 wholes and break them up into thirds.

Create groups of \, \cfrac{2}{3} \, .

There are 18 groups of \, \cfrac{2}{3} \, .

Rashad will have 18 pieces of smaller rope.

6. A recipe calls for 3 \, \cfrac{1}{4} \, cups of strawberries. If Tyler has 5 \, \cfrac{5}{8} \, cups of strawberries, how many will he have left after he makes 1 recipe?

2 \, \cfrac{3}{8} cups

2 \, \cfrac{4}{4} cups

8 \, \cfrac{7}{8} cups

8 \, \cfrac{6}{12} cups

Use the equation 5 \cfrac{5}{8}-3 \cfrac{1}{4}= \, ?

Start with the fraction.

Since \, \cfrac{5}{8} \, and \, \cfrac{1}{4} \, do not have like denominators, the parts are NOT the same size.

A common denominator of 8 can be used.

Multiply the numerator and denominator of \, \cfrac{1}{4} \, by 2 to create an equivalent fraction.

\cfrac{5}{8} \quad and \quad \cfrac{1}{4}=\cfrac{1 \, \times \, 2}{4 \, \times \, 2}=\cfrac{2}{8}

You subtract to see how many parts there are in total: 5-2 = 3.

There are 2 parts and the size is still eighths, so the denominator stays the same.

There will be \, 2 \cfrac{3}{8} \, cups of strawberries left.

Fractions operations FAQs

No, although using these operations will create different denominators and numerators, as long as they are multiplied or divided by the same thing, the value of the fraction will remain the same.

No, unless the question specifies the lowest terms, it is valid to answer without using the least common denominator (LCD). However, as students progress in their understanding of fractions, it is a good idea to encourage them to practice this skill. Also be mindful of standard expectations, as they may vary from state to state.

Yes, just like any other type of number, to solve multistep problems correctly, the order of operations must be followed.

The multiplicative inverse of a number is the reciprocal. For any integer, that is the number written as the numerator over a denominator of 1. For any rational number, that is switching the numerator and denominator.

The next lessons are

• Algebraic expression
• Converting fractions decimals and percents
• Interpret fractions as division
• Fraction word problems

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Fractions Worksheets

Welcome to the fractions worksheets page at Math-Drills.com where the cup is half full! This is one of our more popular pages most likely because learning fractions is incredibly important in a person's life and it is a math topic that many approach with trepidation due to its bad rap over the years. Fractions really aren't that difficult to master especially with the support of our wide selection of worksheets.

This page includes Fractions worksheets for understanding fractions including modeling, comparing, ordering, simplifying and converting fractions and operations with fractions. We start you off with the obvious: modeling fractions. It is a great idea if students can actually understand what a fraction is, so please do spend some time with the modeling aspect. Relating modeling to real life helps a great deal too as it is much easier to relate to half a cookie than to half a square. Ask most students what you get if you add half a cookie and another half a cookie, and they'll probably let you know that it makes one delicious snack.

The other fractions worksheets on this page are devoted to helping students understand the concept of fractions. From comparing and ordering to simplifying and converting... by the time students master the material on this page, operations of fractions will be a walk in the park.

Most Popular Fractions Worksheets this Week

Fraction Circles

Fraction circle manipulatives are mainly used for comparing fractions, but they can be used for a variety of other purposes such as representing and identifying fractions, adding and subtracting fractions, and as probability spinners. There are a variety of options depending on your purpose. Fraction circles come in small and large versions, labeled and unlabeled versions and in three different color schemes: black and white, color, and light gray. The color scheme matches the fraction strips and use colors that are meant to show good contrast among themselves. Do note that there is a significant prevalence of color-blindness in the population, so don't rely on all students being able to differentiate the colors.

Suggested activity for comparing fractions: Photocopy the black and white version onto an overhead projection slide and another copy onto a piece of paper. Alternatively, you can use two pieces of paper and hold them up to the light for this activity. Use a pencil to represent the first fraction on the paper copy. Use a non-permanent overhead pen to represent the second fraction. Lay the slide over the paper and compare the two circles. You should easily be able to tell which is greater or lesser or if the two fractions are equal. Re-use both sheets by erasing the pencil and washing off the marker.

Adding fractions with fraction circles will involve two copies on paper. Cut out the fraction circles and segments of one copy and leave the other copy intact. To add 1/3 + 1/2, for example, place a 1/3 segment and a 1/2 segment into a circle and hold it over various fractions on the intact copy to see what 1/2 + 1/3 is equivalent to. 5/6 or 10/12 should work.

• Small Fraction Circles Small Fraction Circles in Black and White with Labels Small Fraction Circles in Color with Labels Small Fraction Circles in Light Gray with Labels Small Fraction Circles in Black and White Unlabeled Small Fraction Circles in Color Unlabeled Small Fraction Circles in Light Gray Unlabeled
• Large Fraction Circles Large Fraction Circles in Black and White with Labels Large Fraction Circles in Color with Labels Large Fraction Circles in Light Gray with Labels Large Fraction Circles in Black and White Unlabeled Large Fraction Circles in Color Unlabeled Large Fraction Circles in Light Gray Unlabeled

Fraction Strips

Fractions strips are often used for comparing fractions. Students are able to see quite easily the relationships and equivalence between fractions with different denominators. It can be quite useful for students to have two copies: one copy cut into strips and the other copy kept intact. They can then use the cut-out strips on the intact page to individually compare fractions. For example, they can use the halves strip to see what other fractions are equivalent to one-half. This can also be accomplished with a straight edge such as a ruler without cutting out any strips. Pairs or groups of strips can also be compared side-by-side if they are cut out. Addition and subtraction (etc.) are also possibilities; for example, adding a one-quarter and one-third can be accomplished by shifting the thirds strip so that it starts at the end of one-quarter then finding a strip that matches the end of the one-third mark (7/12 should do it).

Teachers might consider copying the fraction strips onto overhead projection acetates for whole class or group activities. Acetate versions are also useful as a hands-on manipulative for students in conjunction with an uncut page.

The "Smart" Fraction Strips include strips in a more useful order, eliminate the 7ths and 11ths strips as they don't have any equivalents and include 15ths and 16ths. The colors are consistent with the classic versions, so the two sets can be combined.

• Classic Fraction Strips with Labels Classic Fraction Strips in Black and White With Labels Classic Fraction Strips in Color With Labels Classic Fraction Strips in Gray With Labels
• Unlabeled Classic Fraction Strips Classic Fraction Strips in Black and White Unlabeled Classic Fraction Strips in Color Unlabeled Classic Fraction Strips in Gray Unlabeled
• Smart Fraction Strips with Labels Smart Fraction Strips in Black and White With Labels Smart Fraction Strips in Color With Labels Smart Fraction Strips in Gray With Labels

Modeling fractions

Fractions can represent parts of a group or parts of a whole. In these worksheets, fractions are modeled as parts of a group. Besides using the worksheets in this section, you can also try some more interesting ways of modeling fractions. Healthy snacks can make great models for fractions. Can you cut a cucumber into thirds? A tomato into quarters? Can you make two-thirds of the grapes red and one-third green?

• Modeling Fractions with Groups of Shapes Coloring Groups of Shapes to Represent Fractions Identifying Fractions from Colored Groups of Shapes (Only Simplified Fractions up to Eighths) Identifying Fractions from Colored Groups of Shapes (Halves Only) Identifying Fractions from Colored Groups of Shapes (Halves and Thirds) Identifying Fractions from Colored Groups of Shapes (Halves, Thirds and Fourths) Identifying Fractions from Colored Groups of Shapes (Up to Fifths) Identifying Fractions from Colored Groups of Shapes (Up to Sixths) Identifying Fractions from Colored Groups of Shapes (Up to Eighths) Identifying Fractions from Colored Groups of Shapes (OLD Version; Print Too Light)
• Modeling Fractions with Rectangles Modeling Halves Modeling Thirds Modeling Halves and Thirds Modeling Fourths (Color Version) Modeling Fourths (Grey Version) Coloring Fourths Models Modeling Fifths Coloring Fifths Models Modeling Sixths Coloring Sixths Models
• Modeling Fractions with Circles Modeling Halves, Thirds and Fourths Coloring Halves, Thirds and Fourths Modeling Halves, Thirds, Fourths, and Fifths Coloring Halves, Thirds, Fourths, and Fifths Modeling Halves to Sixths Coloring Halves to Sixths Modeling Halves to Eighths Coloring Halves to Eighths Modeling Halves to Twelfths Coloring Halves to Twelfths

Ratio and Proportion Worksheets

The equivalent fractions models worksheets include only the "baking fractions" in the A versions. To see more difficult and varied fractions, please choose the B to J versions after loading the A version. More picture ratios can be found on holiday and seasonal pages. Try searching for picture ratios to find more.

• Picture Ratios Autumn Trees Part-to-Part Picture Ratios ( Grouped ) Autumn Trees Part-to-Part and Part-to-Whole Picture Ratios ( Grouped )
• Equivalent Fractions Equivalent Fractions With Blanks ( Multiply Right ) ✎ Equivalent Fractions With Blanks ( Divide Left ) ✎ Equivalent Fractions With Blanks ( Multiply Right or Divide Left ) ✎ Equivalent Fractions With Blanks ( Divide Right ) ✎ Equivalent Fractions With Blanks ( Multiply Left ) ✎ Equivalent Fractions With Blanks ( Multiply Left or Divide Right ) ✎ Equivalent Fractions With Blanks ( Multiply or Divide Right ) ✎ Equivalent Fractions With Blanks ( Multiply or Divide Left ) ✎ Equivalent Fractions With Blanks ( Multiply or Divide in Either Direction ) ✎ Are These Fractions Equivalent? (Multiplier 2 to 5) Are These Fractions Equivalent? (Multiplier 5 to 15) Equivalent Fractions Models Equivalent Fractions Models with the Simplified Fraction First Equivalent Fractions Models with the Simplified Fraction Second
• Equivalent Ratios Equivalent Ratios with Blanks Only on Right Equivalent Ratios with Blanks Anywhere Equivalent Ratios with x 's

Comparing and Ordering Fractions

Comparing fractions involves deciding which of two fractions is greater in value or if the two fractions are equal in value. There are generally four methods that can be used for comparing fractions. First is to use common denominators . If both fractions have the same denominator, comparing the fractions simply involves comparing the numerators. Equivalent fractions can be used to convert one or both fractions, so they have common denominators. A second method is to convert both fractions to a decimal and compare the decimal numbers. Visualization is the third method. Using something like fraction strips , two fractions can be compared with a visual tool. The fourth method is to use a cross-multiplication strategy where the numerator of the first fraction is multiplied by the denominator of the second fraction; then the numerator of the second fraction is multiplied by the denominator of the first fraction. The resulting products can be compared to decide which fraction is greater (or if they are equal).

• Comparing Proper Fractions Comparing Proper Fractions to Sixths ✎ Comparing Proper Fractions to Ninths (No Sevenths) ✎ Comparing Proper Fractions to Ninths ✎ Comparing Proper Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Proper Fractions to Twelfths ✎

The worksheets in this section also include improper fractions. This might make the task of comparing even easier for some questions that involve both a proper and an improper fraction. If students recognize one fraction is greater than one and the other fraction is less than one, the greater fraction will be obvious.

• Comparing Proper and Improper Fractions Comparing Proper and Improper Fractions to Sixths ✎ Comparing Proper and Improper Fractions to Ninths (No Sevenths) ✎ Comparing Proper and Improper Fractions to Ninths ✎ Comparing Proper and Improper Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Proper and Improper Fractions to Twelfths ✎ Comparing Improper Fractions to Sixths ✎ Comparing Improper Fractions to Ninths (No Sevenths) ✎ Comparing Improper Fractions to Ninths ✎ Comparing Improper Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Improper Fractions to Twelfths ✎

This section additionally includes mixed fractions. When comparing mixed and improper fractions, it is useful to convert one of the fractions to the other's form either in writing or mentally. Converting to a mixed fraction is probably the better route since the first step is to compare the whole number portions, and if one is greater than the other, the proper fraction portion can be ignored. If the whole number portions are equal, the proper fractions must be compared to see which number is greater.

• Comparing Proper, Improper and Mixed Fractions Comparing Proper, Improper and Mixed Fractions to Sixths ✎ Comparing Proper, Improper and Mixed Fractions to Ninths (No Sevenths) ✎ Comparing Proper, Improper and Mixed Fractions to Ninths ✎ Comparing Proper, Improper and Mixed Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Proper, Improper and Mixed Fractions to Twelfths ✎
• Comparing Improper and Mixed Fractions Comparing Improper and Mixed Fractions to Sixths ✎ Comparing Improper and Mixed Fractions to Ninths (No Sevenths) ✎ Comparing Improper and Mixed Fractions to Ninths ✎ Comparing Improper and Mixed Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Improper and Mixed Fractions to Twelfths ✎
• Comparing Mixed Fractions Comparing Mixed Fractions to Sixths ✎ Comparing Mixed Fractions to Ninths (No Sevenths) ✎ Comparing Mixed Fractions to Ninths ✎ Comparing Mixed Fractions to Twelfths (No Sevenths; No Elevenths) ✎ Comparing Mixed Fractions to Twelfths ✎

Many of the same strategies that work for comparing fractions also work for ordering fractions. Using manipulatives such as fraction strips, using number lines, or finding decimal equivalents will all have your student(s) putting fractions in the correct order in no time. We've probably said this before, but make sure that you emphasize that when comparing or ordering fractions, students understand that the whole needs to be the same. Comparing half the population of Canada with a third of the population of the United States won't cut it. Try using some visuals to reinforce this important concept. Even though we've included number lines below, feel free to use your own strategies.

• Ordering Fractions with Easy Denominators on a Number Line Ordering Fractions with Easy Denominators to 10 on a Number Line Ordering Fractions with Easy Denominators to 24 on a Number Line Ordering Fractions with Easy Denominators to 60 on a Number Line Ordering Fractions with Easy Denominators to 100 on a Number Line
• Ordering Fractions with Easy Denominators on a Number Line (Including Negative Fractions) Ordering Fractions with Easy Denominators to 10 + Negatives on a Number Line Ordering Fractions with Easy Denominators to 24 + Negatives on a Number Line Ordering Fractions with Easy Denominators to 60 + Negatives on a Number Line Ordering Fractions with Easy Denominators to 100 + Negatives on a Number Line
• Ordering Fractions with All Denominators on a Number Line Ordering Fractions with All Denominators to 10 on a Number Line Ordering Fractions with All Denominators to 24 on a Number Line Ordering Fractions with All Denominators to 60 on a Number Line Ordering Fractions with All Denominators to 100 on a Number Line
• Ordering Fractions with All Denominators on a Number Line (Including Negative Fractions) Ordering Fractions with All Denominators to 10 + Negatives on a Number Line Ordering Fractions with All Denominators to 24 + Negatives on a Number Line Ordering Fractions with All Denominators to 60 + Negatives on a Number Line Ordering Fractions with All Denominators to 100 + Negatives on a Number Line

The ordering fractions worksheets in this section do not include a number line, to allow for students to use various sorting strategies.

• Ordering Positive Fractions Ordering Positive Fractions with Like Denominators Ordering Positive Fractions with Like Numerators Ordering Positive Fractions with Like Numerators or Denominators Ordering Positive Fractions with Proper Fractions Only Ordering Positive Fractions with Improper Fractions Ordering Positive Fractions with Mixed Fractions Ordering Positive Fractions with Improper and Mixed Fractions
• Ordering Positive and Negative Fractions Ordering Positive and Negative Fractions with Like Denominators Ordering Positive and Negative Fractions with Like Numerators Ordering Positive and Negative Fractions with Like Numerators or Denominators Ordering Positive and Negative Fractions with Proper Fractions Only Ordering Positive and Negative Fractions with Improper Fractions Ordering Positive and Negative Fractions with Mixed Fractions Ordering Positive and Negative Fractions with Improper and Mixed Fractions

Simplifying & Converting Fractions Worksheets

Rounding fractions helps students to understand fractions a little better and can be applied to estimating answers to fractions questions. For example, if one had to estimate 1 4/7 × 6, they could probably say the answer was about 9 since 1 4/7 is about 1 1/2 and 1 1/2 × 6 is 9.

• Rounding Fractions with Helper Lines Rounding Fractions to the Nearest Whole with Helper Lines Rounding Mixed Numbers to the Nearest Whole with Helper Lines Rounding Fractions to the Nearest Half with Helper Lines Rounding Mixed Numbers to the Nearest Half with Helper Lines
• Rounding Fractions Rounding Fractions to the Nearest Whole Rounding Mixed Numbers to the Nearest Whole Rounding Fractions to the Nearest Half Rounding Mixed Numbers to the Nearest Half

Learning how to simplify fractions makes a student's life much easier later on when learning operations with fractions. It also helps them to learn that different-looking fractions can be equivalent. One way of demonstrating this is to divide out two equivalent fractions. For example 3/2 and 6/4 both result in a quotient of 1.5 when divided. By practicing simplifying fractions, students will hopefully recognize unsimplified fractions when they start adding, subtracting, multiplying and dividing with fractions.

• Simplifying Fractions Simplify Fractions (easier) Simplify Fractions (harder) Simplify Improper Fractions (easier) Simplify Improper Fractions (harder)
• Converting Between Improper and Mixed Fractions Converting Mixed Fractions to Improper Fractions Converting Improper Fractions to Mixed Fractions Converting Between (both ways) Mixed and Improper Fractions
• Converting Between Fractions and Decimals Converting Fractions to Terminating Decimals Converting Fractions to Terminating and Repeating Decimals Converting Terminating Decimals to Fractions Converting Terminating and Repeating Decimals to Fractions Converting Fractions to Hundredths
• Converting Between Fractions, Decimals, Percents and Ratios with Terminating Decimals Only Converting Fractions to Decimals, Percents and Part-to- Part Ratios ( Terminating Decimals Only) Converting Fractions to Decimals, Percents and Part-to- Whole Ratios ( Terminating Decimals Only) Converting Decimals to Fractions, Percents and Part-to- Part Ratios ( Terminating Decimals Only) Converting Decimals to Fractions, Percents and Part-to- Whole Ratios ( Terminating Decimals Only) Converting Percents to Fractions, Decimals and Part-to- Part Ratios ( Terminating Decimals Only) Converting Percents to Fractions, Decimals and Part-to- Whole Ratios ( Terminating Decimals Only) Converting Part-to-Part Ratios to Fractions, Decimals and Percents ( Terminating Decimals Only) Converting Part-to-Whole Ratios to Fractions, Decimals and Percents ( Terminating Decimals Only) Converting Various Fractions, Decimals, Percents and Part-to- Part Ratios ( Terminating Decimals Only) Converting Various Fractions, Decimals, Percents and Part-to- Whole Ratios ( Terminating Decimals Only)
• Converting Between Fractions, Decimals, Percents and Ratios with Terminating and Repeating Decimals Converting Fractions to Decimals, Percents and Part-to- Part Ratios Converting Fractions to Decimals, Percents and Part-to- Whole Ratios Converting Decimals to Fractions, Percents and Part-to- Part Ratios Converting Decimals to Fractions, Percents and Part-to- Whole Ratios Converting Percents to Fractions, Decimals and Part-to- Part Ratios Converting Percents to Fractions, Decimals and Part-to- Whole Ratios Converting Part-to-Part Ratios to Fractions, Decimals and Percents Converting Part-to-Whole Ratios to Fractions, Decimals and Percents Converting Various Fractions, Decimals, Percents and Part-to- Part Ratios Converting Various Fractions, Decimals, Percents and Part-to- Whole Ratios Converting Various Fractions, Decimals, Percents and Part-to- Part Ratios with 7ths and 11ths Converting Various Fractions, Decimals, Percents and Part-to- Whole Ratios with 7ths and 11ths

Multiplying Fractions

Multiplying fractions is usually less confusing operationally than any other operation and can be less confusing conceptually if approached in the right way. The algorithm for multiplying is simply multiply the numerators then multiply the denominators. The magic word in understanding the multiplication of fractions is, "of." For example what is two-thirds OF six? What is a third OF a half? When you use the word, "of," it gets much easier to visualize fractions multiplication. Example: cut a loaf of bread in half, then cut the half into thirds. One third OF a half loaf of bread is the same as 1/3 x 1/2 and tastes delicious with butter.

• Multiplying Two Proper Fraction Multiplying Two Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ ✎ Multiplying Two Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Proper Fractions with No Simplifying (Printable Only) Multiplying Two Proper Fractions with All Simplifying (Printable Only) Multiplying Two Proper Fractions with Some Simplifying (Printable Only)
• Multiplying Proper and Improper Fractions Multiplying Proper and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Improper Fractions with No Simplifying (Printable Only) Multiplying Proper and Improper Fractions with All Simplifying (Printable Only) Multiplying Proper and Improper Fractions with Some Simplifying (Printable Only)
• Multiplying Two Improper Fractions Multiplying Two Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Improper Fractions with No Simplifying (Printable Only) Multiplying Two Improper Fractions with All Simplifying (Printable Only) Multiplying Two Improper Fractions with Some Simplifying (Printable Only)
• Multiplying Proper and Mixed Fractions Multiplying Proper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper and Mixed Fractions with No Simplifying (Printable Only) Multiplying Proper and Mixed Fractions with All Simplifying (Printable Only) Multiplying Proper and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying Two Mixed Fractions Multiplying Two Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Two Mixed Fractions with No Simplifying (Printable Only) Multiplying Two Mixed Fractions with All Simplifying (Printable Only) Multiplying Two Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying Whole Numbers and Proper Fractions Multiplying Whole Numbers and Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Proper Fractions with No Simplifying (Printable Only) Multiplying Whole Numbers and Proper Fractions with All Simplifying (Printable Only) Multiplying Whole Numbers and Proper Fractions with Some Simplifying (Printable Only)
• Multiplying Whole Numbers and Improper Fractions Multiplying Whole Numbers and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Improper Fractions with No Simplifying (Printable Only) Multiplying Whole Numbers and Improper Fractions with All Simplifying (Printable Only) Multiplying Whole Numbers and Improper Fractions with Some Simplifying (Printable Only)
• Multiplying Whole Numbers and Mixed Fractions Multiplying Whole Numbers and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Whole Numbers and Mixed Fractions with No Simplifying (Printable Only) Multiplying Whole Numbers and Mixed Fractions with All Simplifying (Printable Only) Multiplying Whole Numbers and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying Proper, Improper and Mixed Fractions Multiplying Proper, Improper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper, Improper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper, Improper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Proper, Improper and Mixed Fractions with No Simplifying (Printable Only) Multiplying Proper, Improper and Mixed Fractions with All Simplifying (Printable Only) Multiplying Proper, Improper and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying 3 Fractions Multiplying 3 Proper Fractions (Fillable, Savable, Printable) ✎ Multiplying 3 Proper and Improper Fractions (Fillable, Savable, Printable) ✎ Multiplying Proper and Improper Fractions and Whole Numbers (3 factors) (Fillable, Savable, Printable) ✎ Multiplying Fractions and Mixed Fractions (3 factors) (Fillable, Savable, Printable) ✎ Multiplying 3 Mixed Fractions (Fillable, Savable, Printable) ✎

Dividing Fractions

Conceptually, dividing fractions is probably the most difficult of all the operations, but we're going to help you out. The algorithm for dividing fractions is just like multiplying fractions, but you find the inverse of the second fraction or you cross-multiply. This gets you the right answer which is extremely important especially if you're building a bridge. We told you how to conceptualize fraction multiplication, but how does it work with division? Easy! You just need to learn the magic phrase: "How many ____'s are there in ______? For example, in the question 6 ÷ 1/2, you would ask, "How many halves are there in 6?" It becomes a little more difficult when both numbers are fractions, but it isn't a giant leap to figure it out. 1/2 ÷ 1/4 is a fairly easy example, especially if you think in terms of U.S. or Canadian coins. How many quarters are there in a half dollar?

• Dividing Two Proper Fractions Dividing Two Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Proper Fractions with No Simplifying (Printable Only) Dividing Two Proper Fractions with All Simplifying (Printable Only) Dividing Two Proper Fractions with Some Simplifying (Printable Only)
• Dividing Proper and Improper Fractions Dividing Proper and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Improper Fractions with No Simplifying (Printable Only) Dividing Proper and Improper Fractions with All Simplifying (Printable Only) Dividing Proper and Improper Fractions with Some Simplifying (Printable Only)
• Dividing Two Improper Fractions Dividing Two Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Improper Fractions with No Simplifying (Printable Only) Dividing Two Improper Fractions with All Simplifying (Printable Only) Dividing Two Improper Fractions with Some Simplifying (Printable Only)
• Dividing Proper and Mixed Fractions Dividing Proper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper and Mixed Fractions with No Simplifying (Printable Only) Dividing Proper and Mixed Fractions with All Simplifying (Printable Only) Dividing Proper and Mixed Fractions with Some Simplifying (Printable Only)
• Dividing Two Mixed Fractions Dividing Two Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Two Mixed Fractions with No Simplifying (Printable Only) Dividing Two Mixed Fractions with All Simplifying (Printable Only) Dividing Two Mixed Fractions with Some Simplifying (Printable Only)
• Dividing Whole Numbers and Proper Fractions Dividing Whole Numbers and Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Proper Fractions with No Simplifying (Printable Only) Dividing Whole Numbers and Proper Fractions with All Simplifying (Printable Only) Dividing Whole Numbers and Proper Fractions with Some Simplifying (Printable Only)
• Dividing Whole Numbers and Improper Fractions Dividing Whole Numbers and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Improper Fractions with No Simplifying (Printable Only) Dividing Whole Numbers and Improper Fractions with All Simplifying (Printable Only) Dividing Whole Numbers and Improper Fractions with Some Simplifying (Printable Only)
• Dividing Whole Numbers and Mixed Fractions Dividing Whole Numbers and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Whole Numbers and Mixed Fractions with No Simplifying (Printable Only) Dividing Whole Numbers and Mixed Fractions with All Simplifying (Printable Only) Dividing Whole Numbers and Mixed Fractions with Some Simplifying (Printable Only)
• Dividing Proper, Improper and Mixed Fractions Dividing Proper, Improper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper, Improper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper, Improper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Proper, Improper and Mixed Fractions with No Simplifying (Printable Only) Dividing Proper, Improper and Mixed Fractions with All Simplifying (Printable Only) Dividing Proper, Improper and Mixed Fractions with Some Simplifying (Printable Only)
• Dividing 3 Fractions Dividing 3 Fractions Dividing 3 Fractions (Some Whole Numbers) Dividing 3 Fractions (Some Mixed) Dividing 3 Mixed Fractions

Multiplying and Dividing Fractions

This section includes worksheets with both multiplication and division mixed on each worksheet. Students will have to pay attention to the signs.

• Multiplying and Dividing Two Proper Fractions Multiplying and Dividing Two Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Proper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Two Proper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Two Proper Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Proper and Improper Fractions Multiplying and Dividing Proper and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Improper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Proper and Improper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Proper and Improper Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Two Improper Fractions Multiplying and Dividing Two Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Improper Fractions (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Improper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Two Improper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Two Improper Fractions (Printable Only)
• Multiplying and Dividing Proper and Mixed Fractions Multiplying and Dividing Proper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper and Mixed Fractions with No Simplifying (Printable Only) Multiplying and Dividing Proper and Mixed Fractions with All Simplifying (Printable Only) Multiplying and Dividing Proper and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Two Mixed Fractions Multiplying and Dividing Two Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Two Mixed Fractions with No Simplifying (Printable Only) Multiplying and Dividing Two Mixed Fractions with All Simplifying (Printable Only) Multiplying and Dividing Two Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Whole Numbers and Proper Fractions Fractions Multiplying and Dividing Whole Numbers and Proper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Proper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Proper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Proper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Proper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Proper Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Whole Numbers and Improper Fractions Multiplying and Dividing Whole Numbers and Improper Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Improper Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Improper Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Improper Fractions with No Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Improper Fractions with All Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Improper Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Whole Numbers and Mixed Fractions Multiplying and Dividing Whole Numbers and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Whole Numbers and Mixed Fractions with No Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Mixed Fractions with All Simplifying (Printable Only) Multiplying and Dividing Whole Numbers and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing Proper, Improper and Mixed Fractions Multiplying and Dividing Proper, Improper and Mixed Fractions with No Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper, Improper and Mixed Fractions with All Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper, Improper and Mixed Fractions with Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying and Dividing Proper, Improper and Mixed Fractions with No Simplifying (Printable Only) Multiplying and Dividing Proper, Improper and Mixed Fractions with All Simplifying (Printable Only) Multiplying and Dividing Proper, Improper and Mixed Fractions with Some Simplifying (Printable Only)
• Multiplying and Dividing 3 Fractions Multiplying/Dividing Fractions (three factors) Multiplying/Dividing Mixed Fractions (3 factors)

Adding Fractions

Adding fractions requires the annoying common denominator. Make it easy on your students by first teaching the concepts of equivalent fractions and least common multiples. Once students are familiar with those two concepts, the idea of finding fractions with common denominators for adding becomes that much easier. Spending time on modeling fractions will also help students to understand fractions addition. Relating fractions to familiar examples will certainly help. For example, if you add a 1/2 banana and a 1/2 banana, you get a whole banana. What happens if you add a 1/2 banana and 3/4 of another banana?

• Adding Two Proper Fractions with Equal Denominators and Proper Fraction Results Adding Two Proper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Proper Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Equal Denominators, Proper Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Equal Denominators, Proper Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Equal Denominators and Mixed Fraction Results Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Equal Denominators, Mixed Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Similar Denominators and Proper Fraction Results Adding Two Proper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Proper Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Similar Denominators, Proper Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Similar Denominators, Proper Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Similar Denominators and Mixed Fraction Results Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Similar Denominators, Mixed Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Unlike Denominators and Proper Fraction Results Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Unlike Denominators, Proper Fractions Result, and Some Simplifying (Printable Only)
• Adding Two Proper Fractions with Unlike Denominators and Mixed Fraction Results Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Result, and No Simplifying (Printable Only) Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Result, and All Simplifying (Printable Only) Adding Two Proper Fractions with Unlike Denominators, Mixed Fractions Result, and Some Simplifying (Printable Only)
• Adding Proper and Improper Fractions with Equal Denominators Adding Proper and Improper Fractions with Equal Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Equal Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Equal Denominators and No Simplifying (Printable Only) Adding Proper and Improper Fractions with Equal Denominators and All Simplifying (Printable Only) Adding Proper and Improper Fractions with Equal Denominators and Some Simplifying (Printable Only)
• Adding Proper and Improper Fractions with Similar Denominators Adding Proper and Improper Fractions with Similar Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Similar Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Similar Denominators and No Simplifying (Printable Only) Adding Proper and Improper Fractions with Similar Denominators and All Simplifying (Printable Only) Adding Proper and Improper Fractions with Similar Denominators and Some Simplifying (Printable Only)
• Adding Proper and Improper Fractions with Unlike Denominators Adding Proper and Improper Fractions with Unlike Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Unlike Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Proper and Improper Fractions with Unlike Denominators and No Simplifying (Printable Only) Adding Proper and Improper Fractions with Unlike Denominators and All Simplifying (Printable Only) Adding Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Printable Only)

A common strategy to use when adding mixed fractions is to convert the mixed fractions to improper fractions, complete the addition, then switch back. Another strategy which requires a little less brainpower is to look at the whole numbers and fractions separately. Add the whole numbers first. Add the fractions second. If the resulting fraction is improper, then it needs to be converted to a mixed number. The whole number portion can be added to the original whole number portion.

• Adding Two Mixed Fractions with Equal Denominators Adding Two Mixed Fractions with Equal Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Equal Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Equal Denominators and No Simplifying (Printable Only) Adding Two Mixed Fractions with Equal Denominators and All Simplifying (Printable Only) Adding Two Mixed Fractions with Equal Denominators and Some Simplifying (Printable Only)
• Adding Two Mixed Fractions with Similar Denominators Adding Two Mixed Fractions with Similar Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Similar Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Similar Denominators and Some Simplifying Adding Two Mixed Fractions with Similar Denominators and No Simplifying (Printable Only) Adding Two Mixed Fractions with Similar Denominators and All Simplifying (Printable Only) Adding Two Mixed Fractions with Similar Denominators and Some Simplifying (Printable Only)
• Adding Two Mixed Fractions with Unlike Denominators Adding Two Mixed Fractions with Unlike Denominators and No Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Unlike Denominators and All Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Two Mixed Fractions with Unlike Denominators and No Simplifying (Printable Only) Adding Two Mixed Fractions with Unlike Denominators and All Simplifying (Printable Only) Adding Two Mixed Fractions with Unlike Denominators and Some Simplifying (Printable Only)

Subtracting Fractions

There isn't a lot of difference between adding and subtracting fractions. Both require a common denominator which requires some prerequisite knowledge. The only difference is the second and subsequent numerators are subtracted from the first one. There is a danger that you might end up with a negative number when subtracting fractions, so students might need to learn what it means in that case. When it comes to any concept in fractions, it is always a good idea to relate it to a familiar or easy-to-understand situation. For example, 7/8 - 3/4 = 1/8 could be given meaning in the context of a race. The first runner was 7/8 around the track when the second runner was 3/4 around the track. How far ahead was the first runner? (1/8 of the track).

• Subtracting Two Proper Fractions with Equal Denominators and Proper Fraction Results Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Two Proper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Two Proper Fractions with Similar Denominators and Proper Fraction Results Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Two Proper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Two Proper Fractions with Unlike Denominators and Proper Fraction Results Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Two Proper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Equal Denominators and Proper Fraction Results Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Equal Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Similar Denominators and Proper Fraction Results Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Similar Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Unlike Denominators and Proper Fraction Results Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Unlike Denominators, Proper Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Equal Denominators and Mixed Fraction Results Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Equal Denominators, Mixed Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Similar Denominators and Mixed Fraction Results Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Similar Denominators, Mixed Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Proper and Improper Fractions with Unlike Denominators and Mixed Fraction Results Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and No Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and All Simplifying (Printable Only) Subtracting Proper and Improper Fractions with Unlike Denominators, Mixed Fractions Results, and Some Simplifying (Printable Only)
• Subtracting Mixed Fractions with Equal Denominators Subtracting Mixed Fractions with Equal Denominators, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Equal Denominators, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Equal Denominators, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Equal Denominators, and No Simplifying (Printable Only) Subtracting Mixed Fractions with Equal Denominators, and All Simplifying (Printable Only) Subtracting Mixed Fractions with Equal Denominators, and Some Simplifying (Printable Only)
• Subtracting Mixed Fractions with Similar Denominators Subtracting Mixed Fractions with Similar Denominators, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Similar Denominators, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Similar Denominators, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Similar Denominators, and No Simplifying (Printable Only) Subtracting Mixed Fractions with Similar Denominators, and All Simplifying (Printable Only) Subtracting Mixed Fractions with Similar Denominators, and Some Simplifying (Printable Only)
• Subtracting Mixed Fractions with Unlike Denominators Subtracting Mixed Fractions with Unlike Denominators, and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Unlike Denominators, and All Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Unlike Denominators, and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Mixed Fractions with Unlike Denominators, and No Simplifying (Printable Only) Subtracting Mixed Fractions with Unlike Denominators, and All Simplifying (Printable Only) Subtracting Mixed Fractions with Unlike Denominators, and Some Simplifying (Printable Only)

Adding and Subtracting Fractions

Mixing up the signs on operations with fractions worksheets makes students pay more attention to what they are doing and allows for a good test of their skills in more than one operation.

• Adding and Subtracting Proper and Improper Fractions Adding and Subtracting Proper and Improper Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Proper and Improper Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Proper and Improper Fractions with Equal Denominators and Some Simplifying (Printable Only) Adding and Subtracting Proper and Improper Fractions with Similar Denominators and Some Simplifying (Printable Only) Adding and Subtracting Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Printable Only)
• Adding and Subtracting Mixed Fractions Adding and Subtracting Mixed Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Mixed Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Mixed Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ Adding and Subtracting Mixed Fractions with Equal Denominators and Some Simplifying (Printable Only) Adding and Subtracting Mixed Fractions with Similar Denominators and Some Simplifying (Printable Only) Adding and Subtracting Mixed Fractions with Unlike Denominators and Some Simplifying (Printable Only) Adding/Subtracting Three Fractions/Mixed Fractions

All Operations Fractions Worksheets

• All Operations with Two Proper Fractions with Equal Denominators and Proper Fraction Results All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and No Simplifying (Printable Only) All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and All Simplifying (Printable Only) All Operations with Two Proper Fractions with Equal Denominators, Proper Fractions Results and Some Simplifying (Printable Only)
• All Operations with Two Proper Fractions with Similar Denominators and Proper Fraction Results All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and No Simplifying (Printable Only) All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and All Simplifying (Printable Only) All Operations with Two Proper Fractions with Similar Denominators, Proper Fractions Results and Some Simplifying (Printable Only)
• All Operations with Two Proper Fractions with Unlike Denominators and Proper Fraction Results All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and No Simplifying (Printable Only) All Operations with Two Proper Fractions with Unlike Denominators, Proper Fractions Results and All Simplifying (Printable Only) All Operations with Two Proper Fractions with Unlike Denominators, Mixed Fractions Results and Some Simplifying (Printable Only)
• All Operations with Proper and Improper Fractions with Equal Denominators All Operations with Proper and Improper Fractions with Equal Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Equal Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Equal Denominators and No Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Equal Denominators and All Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Equal Denominators and Some Simplifying (Printable Only)
• All Operations with Proper and Improper Fractions with Similar Denominators All Operations with Proper and Improper Fractions with Similar Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Similar Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Similar Denominators and No Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Similar Denominators and All Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Similar Denominators and Some Simplifying (Printable Only)
• All Operations with Proper and Improper Fractions with Unlike Denominators All Operations with Proper and Improper Fractions with Unlike Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Unlike Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Proper and Improper Fractions with Unlike Denominators and No Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Unlike Denominators and All Simplifying (Printable Only) All Operations with Proper and Improper Fractions with Unlike Denominators and Some Simplifying (Printable Only)
• All Operations with Two Mixed Fractions with Equal Denominators All Operations with Two Mixed Fractions with Equal Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Equal Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Equal Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Equal Denominators and No Simplifying (Printable Only) All Operations with Two Mixed Fractions with Equal Denominators and All Simplifying (Printable Only) All Operations with Two Mixed Fractions with Equal Denominators and Some Simplifying (Printable Only)
• All Operations with Two Mixed Fractions with Similar Denominators All Operations with Two Mixed Fractions with Similar Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Similar Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Similar Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Similar Denominators and No Simplifying (Printable Only) All Operations with Two Mixed Fractions with Similar Denominators and All Simplifying (Printable Only) All Operations with Two Mixed Fractions with Similar Denominators and Some Simplifying (Printable Only)
• All Operations with Two Mixed Fractions with Unlike Denominators All Operations with Two Mixed Fractions with Unlike Denominators and No Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Unlike Denominators and All Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Unlike Denominators and Some Simplifying (Fillable, Savable, Printable) ✎ All Operations with Two Mixed Fractions with Unlike Denominators and No Simplifying (Printable Only) All Operations with Two Mixed Fractions with Unlike Denominators and All Simplifying (Printable Only) All Operations with Two Mixed Fractions with Unlike Denominators and Some Simplifying (Printable Only)
• All Operations with 3 Fractions All Operations with Three Fractions Including Some Improper Fractions All Operations with Three Fractions Including Some Negative and Some Improper Fractions

Operations with Negative Fractions Worksheets

Although some of these worksheets are single operations, it should be helpful to have all of these in the same location. There are some special considerations when completing operations with negative fractions. It is usually very helpful to change any mixed numbers to an improper fraction before proceeding. It is important to pay attention to the signs and know the rules for multiplying positives and negatives (++ = +, +- = -, -+ = - and -- = +).

• Adding with Negative Fractions Adding Negative Proper Fractions with Unlike Denominators Up to Sixths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Negative Proper Fractions with Unlike Denominators Up to Twelfths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Adding Negative Mixed Fractions with Unlike Denominators Up to Sixths, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Adding Negative Mixed Fractions with Unlike Denominators Up to Twelfths, Proper Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Adding Negative Proper Fractions with Denominators Up to Sixths, Proper Fraction Results and Some Simplifying (Printable Only) Adding Negative Proper Fractions with Denominators Up to Twelfths, Proper Fraction Results and Some Simplifying (Printable Only) Adding Negative Mixed Fractions with Denominators Up to Sixths and Some Simplifying (Printable Only) Adding Negative Mixed Fractions with Denominators Up to Twelfths and Some Simplifying (Printable Only)
• Subtracting with Negative Fractions Subtracting Negative Proper Fractions with Unlike Denominators Up to Sixths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Negative Proper Fractions with Unlike Denominators Up to Twelfths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Subtracting Negative Mixed Fractions with Unlike Denominators Up to Sixths, Mixed Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Negative Mixed Fractions with Unlike Denominators Up to Twelfths, Mixed Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Subtracting Negative Proper Fractions with Denominators Up to Sixths, Proper Fraction Results and Some Simplifying (Printable Only) Subtracting Negative Proper Fractions with Denominators Up to Twelfths, Proper Fraction Results and Some Simplifying (Printable Only) Subtracting Negative Mixed Fractions with Denominators Up to Sixths and Some Simplifying (Printable Only) Subtracting Negative Mixed Fractions with Denominators Up to Twelfths and Some Simplifying (Printable Only)
• Multiplying with Negative Fractions Multiplying Negative Proper Fractions with Denominators Up to Sixths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Negative Proper Fractions with Denominators Up to Twelfths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Negative Mixed Fractions with Denominators Up to Sixths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Negative Mixed Fractions with Denominators Up to Twelfths, Proper Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Multiplying Negative Proper Fractions with Denominators Up to Sixths, Proper Fraction Results and Some Simplifying (Printable Only) Multiplying Negative Proper Fractions with Denominators Up to Twelfths, Proper Fraction Results and Some Simplifying (Printable Only) Multiplying Negative Mixed Fractions with Denominators Up to Sixths and Some Simplifying (Printable Only) Multiplying Negative Mixed Fractions with Denominators Up to Twelfths and Some Simplifying (Printable Only)
• Dividing with Negative Fractions Dividing Negative Proper Fractions with Denominators Up to Sixths, Mixed Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Negative Proper Fractions with Denominators Up to Twelfths, Mixed Fractions Results and Some Simplifying (Fillable, Savable, Printable) ✎ Dividing Negative Mixed Fractions with Denominators Up to Twelfths, Mixed Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Dividing Negative Mixed Fractions with Denominators Up to Twelfths, Mixed Fractions Results and No Simplifying (Fillable, Savable, Printable) ✎ Dividing Negative Proper Fractions with Denominators Up to Sixths, Proper Fraction Results and Some Simplifying (Printable Only) Dividing Negative Proper Fractions with Denominators Up to Twelfths, Proper Fraction Results and Some Simplifying (Printable Only) Dividing Negative Mixed Fractions with Denominators Up to Sixths and Some Simplifying (Printable Only) Dividing Negative Mixed Fractions with Denominators Up to Twelfths and Some Simplifying (Printable Only)

Order of Operations with Fractions Worksheets

The order of operations worksheets in this section actually reside on the Order of Operations page, but they are included here for your convenience.

• Order of Operations with Fractions 2-Step Order of Operations with Fractions 3-Step Order of Operations with Fractions 4-Step Order of Operations with Fractions 5-Step Order of Operations with Fractions 6-Step Order of Operations with Fractions
• Order of Operations with Fractions (No Exponents) 2-Step Order of Operations with Fractions (No Exponents) 3-Step Order of Operations with Fractions (No Exponents) 4-Step Order of Operations with Fractions (No Exponents) 5-Step Order of Operations with Fractions (No Exponents) 6-Step Order of Operations with Fractions (No Exponents)
• Order of Operations with Positive and Negative Fractions 2-Step Order of Operations with Positive & Negative Fractions 3-Step Order of Operations with Positive & Negative Fractions 4-Step Order of Operations with Positive & Negative Fractions 5-Step Order of Operations with Positive & Negative Fractions 6-Step Order of Operations with Positive & Negative Fractions

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Adding and subtracting fractions and mixed numbers.

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Add and subtract fractions and mixed numbers with unlike denominators in number and word problems.

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Lectures, Courses, Problem Examples

Operations with Fractions

Doing operations with fractions means applying the four basic mathematical operations:  addition , subtraction , multiplication and division .

Adding and subtracting fractions with the same denominator

Adding and subtracting fraction with the same denominator is easy. All you have to do is add or subtract the numerator while the denominator remains as it is. If the fraction that we get can be reduced to lowest terms then it’s better to do so.

Exam ple 1:  Add the fractions $ \displaystyle \frac{1}{9}+\frac{2}{9}$

Since the denominator is the same we add the numerators and keep the same denominator

$\displaystyle \frac{1}{9}+\frac{2}{9}=\frac{{1+2}}{9}=\frac{3}{9}$

The we reduce the sum to the lowest term

$\displaystyle \frac{3}{9}=\frac{3}{{3\times 3}}=\frac{1}{3}$

So $\displaystyle \frac{1}{9}+\frac{2}{9}=\frac{3}{9}=\frac{1}{3}$

Example 2 : Subtract the fractions $ \displaystyle \frac{5}{7}-\frac{3}{7}$

Solution : Since the denominator is the same we substract the numerators and keep the same denominator

$ \displaystyle \frac{5}{7}-\frac{3}{7}=\frac{{5-3}}{7}=\frac{2}{7}$

So, $\displaystyle \frac{5}{7}-\frac{3}{7}=\frac{2}{7}$

Adding and subtracting fractions with different denominator

When adding and subtracting fraction with different denominators you have to turn the fractions into equivalent fractions by finding the least common denominator.Then reduce the sum into the lowest term if possible.

Example 3: Add the fractions $ \displaystyle \frac{2}{5}+\frac{3}{{15}}$

Since the fractions have diffrent denominator we find the LCD

LCD(5, 15) = 3 x 5 = 15

Converting the fractions into  equivalent fractions    with 15 as a denominator.

$\displaystyle \frac{2}{5}\times \frac{3}{3}=\frac{6}{{15}}$

$\displaystyle \frac{3}{{15}}$ is a fraction with the denominator 15 so no need to be converted.

Now since our fractions have similar denominators we add the numerators, the denominator stays the same and then we reduce the sum.

$\displaystyle \frac{6}{{15}}+\frac{3}{{15}}=\frac{9}{{15}}=\frac{3}{5}$

So, $ \displaystyle \frac{2}{5}+\frac{3}{{15}}=\frac{3}{5}$

Example 4 : Subtract the fraction $ \displaystyle \frac{1}{{12}}-\frac{2}{9}$

We find the LCD since we have different denominators

12 = 2 x 2 x 3

LCD(9, 12) = 2 x 2 x 3 x 3 = 36

Converting the fractions into equivalent fractions with 36 as a denominator.

$ \displaystyle \frac{1}{{12}}\times \frac{3}{3}=\frac{3}{{36}}$

$\displaystyle \frac{2}{9}\times \frac{4}{4}=\frac{8}{{36}}$

Now since the fractions have similar denominators we substract the numerators, the denominator stays the same.

$ \displaystyle \frac{3}{{36}}-\frac{8}{{36}}=-\frac{5}{{36}}$.

So,$ \displaystyle \frac{1}{{12}}-\frac{2}{9}=-\frac{5}{{36}}$

Adding and subtracting a fraction with a mixed one or two mixed fractions

• Convert the mixed fraction into a improper fraction.
• If the proper fractions have unlike denominators then find the LCD and change the fractions into equivalent one.
• Subtract or add the numerators and leave the dominator as it is.
• Reduce to lowest term if possible.

Example 5: Add the fractions $\displaystyle \frac{6}{7}+2\frac{3}{5}$

Firstly we convert the mixed fraction $\displaystyle 2\frac{3}{5}$ into a improper one .

The whole number is 2 and the denominator is 5, so 5 x 2 = 10

We add the product 10 to the numerator of our fraction: 3+10=13

Since the denominator stays the same our improper fraction is: $\displaystyle \frac{{13}}{5}$

The fractions $\displaystyle\frac{6}{7}$ and $\displaystyle \frac{{13}}{5}$ have diffrent denominators, so we find the LCD to turn them into equivalent one´s.

LCD(7, 5) = 35

$ \displaystyle \frac{6}{7}\times \frac{5}{5}=\frac{{30}}{{35}}$

$\displaystyle \frac{{13}}{5}\times \frac{7}{7}=\frac{{91}}{{35}}$

Add the numerators and leave the denominator as it is :$\displaystyle \frac{{30}}{{35}}+\frac{{91}}{{35}}=\frac{{121}}{{35}}$

So, $\displaystyle \frac{6}{7}+2\frac{3}{5}=\frac{{121}}{{35}}$

Example 6: Subtract the mixed fractions $ \displaystyle 3\frac{2}{7}-2\frac{3}{4}$

Firstly convert the mixed fractions into improper one.

$\displaystyle 2\frac{3}{4}=\frac{{11}}{4}$ and $\displaystyle 3\frac{2}{7}=\frac{{23}}{7}$

Since they have different denominator we change them into equaivalent fractions by finding the LCD.

LCD(4,7)=28

$ \displaystyle \frac{{11}}{4}\times \frac{7}{7}=\frac{{77}}{{28}}$

$\displaystyle \frac{{23}}{7}\times \frac{4}{4}=\frac{{92}}{{28}}$

Subtract the numerators and leave the denominator as it is:

$\displaystyle \frac{{92}}{{28}}-\frac{{77}}{{28}}=\frac{{15}}{{28}}$

So,  $ \displaystyle 3\frac{2}{7}-2\frac{3}{4}=$

$ \displaystyle \frac{{92}}{{28}}-\frac{{77}}{{28}}=\frac{{15}}{{28}}$

Adding and Subtracting Fractions by Using Formulas

There is an easy way to add and subtract fractions without having to find the LCD. This method consist in using formulas that involves cross multiplication of the fractions.

Adding  Fraction

The formula for adding fractions is:

$\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{{ad+bc}}{{bd}}$

Subtracting Fraction

The formula for subtracting fractions is:

$\displaystyle \frac{a}{b}-\frac{c}{d}=\frac{{ad-cd}}{{bd}}$

Example 7:  Add $ \displaystyle \frac{5}{9}+\frac{2}{5}$by using the formula above.

Solution : $\displaystyle \frac{5}{9}+\frac{2}{5}=\frac{{(5\times 5)+(9\times 2)}}{{9\times 5}}=\frac{{25+18}}{{45}}=\frac{{43}}{{45}}$

Example 8:  Subtract$\displaystyle \frac{2}{3}-\frac{3}{9}$ by using the formula above.

Solution : $\displaystyle \frac{2}{3}-\frac{3}{9}=\frac{{(2\times 9)-(3\times 3)}}{{3\times 9}}=\frac{{18-9}}{{27}}=\frac{9}{{27}}=\frac{1}{3}$

Multiplying and Dividing Fractions

$ \displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)$

$ \displaystyle n(A\cap B)=n(A)+n(B)-n(A\cup B)$

$ \displaystyle n(A\cap B)=12+10-18=4$

$ \displaystyle n(A/B)=n(A)-n(A\cap B)$

$ \displaystyle n(A/B)=12-4=8$

Multiplying Fractions Multiplying fractions is even easier than adding and substracting them.All you have to do is multiply the numerators  and the denominators of the fraction then reduce the fraction if possible.If mixed fractions are involved don`t forget to turn them to improper fractions.

The formula for multiplying fractions is:  $\displaystyle \frac{a}{b}\times \frac{c}{d}=\frac{{ac}}{{bd}}$

Example 9:  Multiply $\displaystyle \frac{2}{9}\times \frac{3}{5}$.

Solution:   $\displaystyle \frac{2}{9}\times \frac{3}{5}=\frac{{(2\times 3)}}{{(9\times 5)}}=\frac{6}{{45}}=\frac{2}{{15}}$

Example 10:  Multiply $\displaystyle 2\frac{1}{7}\times \frac{3}{8}$.

Solution:  Firstly we have to turn the mixed fraction into a improper one.

$\displaystyle 2\frac{1}{7}=\frac{{15}}{7}$

Then we multiply by using the formula above.

$\displaystyle 2\frac{1}{7}\times \frac{3}{8}=\frac{{15}}{7}\times \frac{3}{8}=\frac{{45}}{{56}}$

Dividing Fractions

Dividing two fractions is the same as multiplying the first fraction with the reciprocal of the second fraction. So all you have to do is keep the first fraction, change the division sign to multiplication and then flip the second fraction. Then we use the multiplication formula. Don´t forget to reduce the fraction if possible. If it´s easier for you, you can apply directly the formula for dividing fractions.

The formula for dividing fractions is:  $ \displaystyle \frac{a}{b}\div \frac{c}{d}=\frac{{ad}}{{bc}}$

Example 11:  Divide $\displaystyle \frac{2}{4}\div \frac{5}{9}$.

Three simply steps:

Keep the first fraction, flip the second one and then multiply them.

$\displaystyle \frac{2}{4}\div \frac{5}{9}=\frac{2}{4}\times \frac{9}{5}=\frac{{18}}{{20}}=\frac{9}{{10}}$

Using the formula

$ \displaystyle \frac{2}{4}\div \frac{5}{9}=\frac{{(2\times 9)}}{{(4\times 5)}}=\frac{{18}}{{20}}=\frac{9}{{10}}$

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Fractions and their Operations

Table of contents.

September 16, 2020     

Read Time: 5 minutes

A Whole and its Parts

Martin had two equal-sized chocolate bars with different placement of grooves such that he could divide one of them into four pieces and the other one into just two pieces. He gave one piece to each of his six friends. Each one kept wondering if they got an equal share. To answer such questions, fractions come into play! A fraction is a number representing a part of a whole or a group of objects. Let us understand fractions, in an interesting and easy manner, through this article.

What are Fractions?

You have a part of something but not the complete thing.  If I ask you how much do you have, what will be your answer? Take for example the below scenario:

• You ate \(2\) slices of pizza that has a total of \(8\) slices
• You have \(1\) piece from a cake that has a total of \(6\) pieces

In both the above cases, you don’t have the complete object but only a part of it. In other words, you have a fraction of the object. 

When any object is divided into equal numbers of parts and you pick up one or more of those parts, you have a fraction of the object.  Now one very important characteristic of these parts of a fraction is that:  Each part of a fraction has to be equal to other parts. 

Next, let us understand how to represent a fraction. A fraction is represented by a combination of \(2\) numbers arranged one above the other separated by a horizontal dash. The number above the separating line is called a Numerator and the number below is called a Denominator. 

The numerator represents the concerned portion (which may be the parts you have or the parts you don’t have depending on your concern or query) and the Denominator represents the total number of parts or the divisions of the whole or total number of objects in the group. For example, if you ate \(2\) slices out of \(4\) equal slices, you can say that you ate \(\begin{align}\frac{2}{4}\end{align}\) of the pizza or half of the pizza. This means that you have \(2\) equal parts out of \(4\) equal parts. The \(4\) equal parts together make up one object. So this means that you don’t have one complete object and neither do you have none of the objects. You have more than zero but less than one. That is you have a fraction of something.  Similarly, if you eat one more slice out of these, then you have eaten \(\begin{align}\frac{3}{4}\end{align}\) of the pizza and  

\(\begin{align}\frac{1}{4}\end{align}\)  of the pizza is left or one quarter is left.

Types Of Fractions:

There are \(3\) types of fractions: Depending on I eat a part of a pizza or a full pizza along with a part of another pizza, the fractions may be represented in different forms.

Proper fractions: When the Numerator is less than the Denominator, it is called as a Proper fraction Eg. \(\begin{align}\frac{2}{5}, \frac{1}{3}\end{align}\) Improper Fractions: When the Numerator is more than the Denominator, it is called as an Improper fraction Eg. \(\begin{align}\frac{7}{5}, \frac{8}{3}\end{align}\) Mixed fractions: A combination of a Whole number and a Proper fraction is called as a mixed fraction Eg. \(\begin{align}2\frac{2}{5},\,\, 3\frac{2}{3}\end{align}\)

Mathematical Operations On Fractions

Doing calculations on fractions is different than doing the same on the whole numbers. So let us try to understand the steps for performing the various mathematical operations on fractions in detail.

Adding Fractions

Before understanding how to add fractions, let us revise what numerator and denominator represent. The Denominator represents the total number of equal parts of an object. So suppose I ask you to add   \(\begin{align}\frac{2}{8}+\frac{3}{8}\end{align}\)  how will you do it?  Let us imagine a pizza with a total of \(8\) slices. \(\begin{align}\frac{2}{8}\end{align}\) represents \(2\) slices out of \(8\) and \(\begin{align}\frac{3}{8}\end{align}\) represents \(3\) slices out of \(8\) . So if you add these you will have \(5\) slices out of \(8\) . Hence \(\begin{align}\frac{2}{8}+\frac{3}{8}=(\frac{2+3}{8})=\frac{5}{8}\end{align}\) This is fairly simple. To add two or more fractions that have the same denominator, you will have to simply add the values in the numerator while keeping the denominator the same. 

But what if the fractions have different denominators? This is a bit tricky but very simple if you understand the concept. To add two or more fractions that have different denominators, you will first need to make the denominator across all fractions the same by finding the Least Common Multiple (LCM) of the denominators. 

Once you have found the LCM, for each fraction, multiply the denominator with the multiplier so that the denominator becomes the LCM. At the same time, to keep the value of fraction constant, you also need to convert them into equivalent fractions by multiplying the numerator of the fraction with the same multiplier. 

You repeat this operation for all the fractions so that all the fractions have the same denominator and then you can add the fractions easily.  Let us understand with an example. Suppose we want to add \(\begin{align}\frac{2}{5}+\frac{3}{4}\end{align}\) . Since these fractions have different denominators, we cannot simply add the numerators.

First, we have to convert the fractions so that they have the same denominator. The first step to do this is finding the LCM of \(5\) and \(4\) . The LCM of \(5\) and \(4\) is \(20\) . So we will have to convert the fraction so that the denominator of the fraction becomes \(20\) . 

And at the same time, we don’t want to disturb the value of the fraction so we will multiply the individual numerators also with the respective multipliers.  Solve: \(\begin{align}\frac{2}{5}+\frac{3}{4}\end{align}\)

\(\begin{align}&=\frac{2\times 4}{5\times 4}+\frac{3\times 5}{4\times 5}\\&=\frac{8}{20}+\frac{15}{20}\\&=\frac{23}{20}\end{align}\)

Hence \(\begin{align}(\frac{2}{5}+\frac{3}{4})=\frac{23}{20}\end{align}\)

Subtracting Fractions

Now since the addition of fractions is quite clear, subtracting fractions is very similar.  If you want to subtract fractions with the same denominator simply subtract the numerator while keeping the denominator the same. Refer to the example below:  Solve: \(\begin{align}\frac{5}{7}-\frac{1}{7}\end{align}\)

\(\begin{align}&=\frac{(5-1)}{7}\\ &=\frac{4}{7}\end{align}\) Hence \(\begin{align}\frac{5}{7}-\frac{1}{7}=\frac{4}{7}\end{align}\) If you want to subtract the fractions with different denominators, follow the same steps that we did for adding the fractions, and in the last step, subtract the numerators. Refer to the example below: Solve: \(\begin{align}\frac{4}{5}-\frac{1}{2}\end{align}\) \(\begin{align}&= \frac{{(4 \times 2)}}{{(5 \times 2)}} - \frac{{(1 \times 5)}}{{(2 \times 5)}}\\ &= \frac{8}{{10}} - \frac{5}{{10}}\\& = \frac{{8 - 5}}{{10}}\\ &= \frac{3}{{10}}\end{align}\) Hence \(\begin{align}\frac{4}{5}-\frac{1}{2}=\frac{3}{10}\end{align}\)

Multiplying Fractions

Multiplying fractions is fairly simple. When multiplying two or more fractions, the numerator of the solution will be the product of numerators of all fractions and the denominator will be the multiplication result or product of denominators of all the fractions. Refer to the example below:  Solve: \(\begin{align}\frac{3}{4}\times\frac{4}{7}\end{align}\) \(\begin{align}&= \frac{{3 \times 4}}{{5 \times 7}}\\ &= \frac{{12}}{{35}}\end{align}\) Hence \(\begin{align}(\frac{3}{5}\times\frac{4}{7})=\frac{12}{35}\end{align}\)

Dividing Fractions

The method to divide fractions follows the same steps as multiplying fractions but with only one additional step. This additional step is to convert the divide fraction operation into multiplication operation. This is done by exchanging the numerator and denominator of the second fraction in the divide operation.  Take for example the below example: 

Solve: \(\begin{align}\frac{2}{5}\div\frac{5}{8}\end{align}\) To solve this example, we will  Replace the numerator and denominator of the second fraction ( \(\begin{align}\frac{5}{8}\end{align}\) will become \(\begin{align}\frac{8}{5}\end{align}\) ) Replacing the division symbol with multiplication  Solving the equation as a multiplication operation \(\begin{align}\frac{2}{5}\div\frac{5}{8}\end{align}\) \(\begin{align}  &= \frac{{2 \times 8}}{{5 \times 5}}\\  &= \frac{{16}}{{55}} \end{align}\) Hence \(\begin{align}\frac{2}{5}\div\frac{5}{8}=\frac{16}{25}\end{align}\)

Fractions are a very interesting concept in mathematics and lay a solid foundation for advanced levels in mathematics. Fractions also have a wide range of real-world applications. Once you become well versed with fractions, you can solve many real-life examples from daily life.

Home / United States / Math Classes / 6th Grade Math / Operations On Fractions

Operations on Fractions

A fraction is used to represent a whole number that is divided into equal parts. We can perform math operations in fract ions just like we do with whole numbers. Learn the steps involved in adding, subtracting, and multiplying fractions with the help of some examples. ...Read More Read Less

About Operations on Fractions

Addition and Subtraction of Fractions

Multiplication of fractions, example of multiplication of two fractions, example of multiplication of fraction with mixed number, division of fractions, examples of fraction divided by another fraction, examples of a whole number divided by a fraction, real life modeling questions.

• Frequently Asked Questions

There are four basic operations that can be performed on numbers and they are addition, subtraction, multiplication and division. We have learnt how to perform these operations on numbers since junior grades.

For example:

Addition: \( 3+2=5 \)

Subtraction: \( 7-4=3 \)

Multiplication:  \( 5\times 3=15 \)

Division: \( \frac{25}{5}=5 \)

We can perform similar operations on fractions as well. Let us learn more about it.

To add and subtract fractions we use equivalent fractions to write the fractions with a common denominator. To find a suitable common denominator we first find the LCM (least common multiple) of fractions with uncommon denominators. Let us understand this by solving a few examples:

Example 1: Find \( \frac{1}{8}+\frac{3}{5} \) .

Let’s use equivalent fractions to rewrite the fractions with a common denominator.

8 is not a multiple of 5 , so let us rewrite each fraction with a denominator which is LCM of 8 and 5 which can be found as :

\( 8 \times 5=40 \)

\( \frac{1}{8} + \frac{3}{5} = \frac{5}{40} + \frac{24}{40}\)      Rewrite \( \frac{1}{8} \)  as \( \frac{1\times 5}{8\times 5} = \frac{5}{40} \)  and  \( \frac{3}{5} \)  as \( \frac{3\times 8}{5\times 8} = \frac{24}{40} \)  

           \( =\frac{5+24}{40} \)            Solve 5 + 24

           \( =\frac{29}{40} \)

Example 2: Find \( \frac{6}{7}-\frac{3}{4} \).

7 is not a multiple of 4 , so let us rewrite each fraction with a denominator which is LCM of 7 and 4 which can be found as :

\( 7\times 4=28 \)

\( \frac{6}{7}-\frac{3}{4}=\frac{24}{28}-\frac{21}{28} \)        Rewrite  \( \frac{6}{7} \) as \( \frac{6\times 4}{7\times 4}=\frac{24}{28} \)  and  \( \frac{3}{4} \) as \( \frac{3\times 7}{4\times 7}=\frac{21}{28} \)  

           \( =\frac{24-21}{28} \)            Solve 24 – 21

\(~~~~~~~~~~ =\frac{3}{28} \)

To perform multiplication of fractions is an easy task. All we need to do is multiply the numerator to the numerator and denominator to the denominator. Solving a few examples will help you understand the method of multiplying fractions.

Find:  \( \frac{1}{7}\times \frac{3}{4} \)   

\( \frac{1}{7}\times \frac{3}{4}=\frac{1\times 3}{7\times 4} \)      Multiply the numerator to the numerator and denominator to the denominator.

           \( =\frac{3}{28} \)  

Find: \( \frac{2}{3}\times 5\frac{3}{4} \)

\( \frac{2}{3}\times 5\frac{3}{4}=\frac{2}{3}\times \frac{23}{4} \)            Write \( 5\frac{3}{4}\)  as the improper fraction \( \frac{23}{4}\).

             \( =\frac{2\times 23}{3\times 4}\)                Multiply the numerator to the numerator and denominator to the denominator.

             \( =\frac{46}{12}\) or \( \frac{23}{6}\)          Simplify.

When we divide a fraction by another, we need to multiply the dividend with the reciprocal of the divisor. We can observe the method of writing the reciprocal with this explanation.

Reciprocal of \( \frac{a}{b}\) is \( \frac{b}{a}\)

Reciprocal of \( \frac{3}{7}\) is \( \frac{7}{3}\) and Reciprocal of \( \frac{9}{13}\) is \( \frac{13}{9}\)

If we notice closely the product of the fraction and its reciprocal is 1.

For example :

\( \frac{3}{7}\times \frac{7}{3}=1 \) and \( \frac{9}{13}\times \frac{13}{9}=1 \).

Note: Reciprocals are also called multiplicative inverses. 

Now, let us see how reciprocals are used to perform division of fractions. Here are some examples.

Find: \( \frac{1}{5}\div \frac{1}{9} \)

\( \frac{1}{5}\div \frac{1}{9}=\frac{1}{5}\times \frac{9}{1} \)          Multiply by the reciprocal of \( \frac{1}{9} \) which is \( \frac{9}{1} \)

          \( =\frac{1\times 9}{5\times 1} \)              Multiply the fractions.

          \( =\frac{9}{5} \)                  Simplify.

Find: \( \frac{4}{7}\div 5 \)

\( \frac{4}{7}\div 5=\frac{4}{7}\times \frac{1}{5} \)          Multiply by the reciprocal of 5 which is \( \frac{1}{5} \)

          \( =\frac{4\times 1}{7\times 5} \)              Multiply the fractions.

          \( =\frac{4}{35} \)                Simplify.

You have \( \frac{4}{5} \) of a pizza. You divide the remaining pizza in 5 equal parts to distribute among your family members.  What portion of the original pizza will each member get?

We need to divide \( \frac{4}{5} \) into five equal parts.

\( \frac{4}{5}\div 5= \frac{4}{5}\times \frac{1}{5} \)          Multiply by the reciprocal of 5 which is \( \frac{1}{5} \)

          \( =\frac{4\times 1}{5\times 5} \)              Multiply the fractions.

          \( =\frac{4}{25} \)                  Simplify.

Hence, each family member gets \( \frac{4}{25} \)  of the pizza.

What is the reciprocal of a whole number?

As every whole number can be divided by 1 and it won’t change its value, so reciprocal of a whole number like 4 can be written as \( \frac{1}{4} \).

What is the reciprocal of zero?

Reciprocal of zero is not defined as we cannot divide any number with zero.

0 can be written as \( \frac{0}{1} \) but, \( \frac{1}{0} \) is not defined.

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Operations with Fractions

Introduction.

Fractions are an essential part of mathematics that represent a part of a whole. They are used in many real-life situations, such as cooking, sharing resources, and measurements. Understanding how to perform operations with fractions is crucial in solving math problems and working with fractions in everyday life.

In this blog post, we will cover the basics of operations with fractions, including addition, subtraction, multiplication, and division. We will also provide examples and tips to help you master these operations and simplify fractions.

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Adding and Subtracting Fractions

For adding or subtracting fractions, a common denominator is necessary. The common denominator is the same number that both fractions can be written with. Here’s how you can add or subtract fractions:

• Determine the smallest common multiple (LCM) of the denominators.
• Convert each fraction to an equivalent fraction with the LCM as the denominator.
• Combine or deduct the numerators and express the result with the common denominator.

Let’s work through an example problem:

Suppose we need to subtract 3/7 from 4/9.

Step 1: Determine the least common multiple (LCM) of 7 and 9.

The numbers that are divisible by 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70…

The numbers that are divisible by 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90…

63 is the least common multiple (LCM) of 7 and 9.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 63.

4/9 x 7/7 = 28/63

3/7 x 9/9 = 27/63

Step 3: Subtract the numerators and write the result over the common denominator.

28/63 – 27/63 = 1/63

Therefore, 4/9 – 3/7 = 1/63.

By following these steps, we can successfully subtract fractions with different denominators.

Reduce a fraction to its lowest terms with Fraction Simplifier

Multiplying and Dividing Fractions

Multiplying fractions is straightforward. You simply multiply the numerators and denominators separately and write the result as a fraction. Here’s how:

• Multiply the numerators of the fractions.
• Multiply the denominators of the fractions.
• Write the result as a fraction.

Let’s take a different example:

Suppose we need to multiply 3/4 by 2/5.

3/4 x 2/5=(3×2)÷(4×5)= 6/20.

To simplify the result, we need to find the greatest common factor (GCF) of 6 and 20, which is 2. Then, we divide both the numerator and denominator by 2 to get the simplified fraction:

6/20 = 3/10.

Therefore, 3/4 multiplied by 2/5 is equal to 3/10.

By following these steps, we can successfully multiply fractions and simplify the result to its lowest terms.

Dividing fractions is a little more complicated than multiplying fractions. Here’s how:

• Invert the second fraction (the one you’re dividing by).
• Multiply the first fraction by the inverted second fraction.
• Simplify the result if possible.

Let’s consider an example:

Divide 2/3 by 4/5.

2/3 ÷ 4/5 = 2/3 x 5/4 = (2 x 5) / (3 x 4) = 10/12.

To simplify the result, we need to find the GCF of 10 and 12, which is 2. Then, we divide both the numerator and denominator by 2 to get the simplified fraction:

10/12 = 5/6.

Tips for Simplifying Fractions

Simplifying fractions is important because it makes them easier to work with and understand. Here are some tips to help you simplify fractions:

• Find the Greatest Common Factor (GCF) of the numerator and denominator and divide both by it.
• Look for common factors in the numerator and denominator and cancel them out.
• If the numerator and denominator have no common factors, the fraction is already in its simplest form.
• If the numerator is larger than the denominator, the fraction can be written as a mixed number. For example, 7/4 can be written as 1 3/4.
• If the denominator is a multiple of 10, the fraction can be converted to a decimal by dividing the numerator by the denominator.

Simplify the fraction 12/24.

Step 1: Find the GCF of 12 and 24, which is 12.

Step 2: Divide the numerator and denominator by a common factor of 12.

12/24 = 1/2.

Therefore, 12/24 simplifies to 1/2.

Solve multiple types of fraction math problems with the Fractions Calculator

Operations with fractions are an essential part of mathematics that are used in many real-life situations. Understanding how to perform these operations is crucial in solving math problems and working with fractions in everyday life. Adding, subtracting, multiplying, and dividing fractions can be easily mastered with practice and the use of common techniques like finding the common denominator, inverting the second fraction, and simplifying fractions. Simplifying fractions is important because it makes them easier to work with and understand. With these tips and techniques, you can easily perform operations with fractions and simplify them to their simplest form.

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Basic Arithmetic : Operations with Fractions

Study concepts, example questions & explanations for basic arithmetic, all basic arithmetic resources, example questions, example question #1 : operations with fractions.

Multiply the numerators together, then multiply the denominators together.

First, find out how many girls and boys are in the class.

Then, subtract the number of girls from the total number of students in the class to find the number of boys.

Now, we can figure out how many girls enjoy eating strawberries. Multiply the total number of girls by the fraction of girls who enjoy eating strawberries.

Now, do the same with the boys.

Add these two numbers together to get the total number of students who enjoy eating strawberries.

Solve and simplify. 

When multiplying fractions, you simply multiply the numerators and then multiply the denominators. Then simplify the fraction.

Evaluate the following:

This problem involves order of operations.  

The correct order is:  Parenthesis, Exponents, Multiplication, Division, Addition, and Subtraction.  

Rewrite the exponent and cancel out the 4 on the numerator and denominator.

Example Question #1 : Multiplication With Fractions

Please choose the best answer for the question below. 

To do this problem, simply multiply through (i.e. numerator by numerator and denominator by denominator):

Example Question #6 : Multiplication With Fractions

Multiply these fractions:

When multiplying fractions, all we have to do is multiply the numerators together and multiply the denominators together:

Simply the fraction to get the final answer:

Example Question #10 : Multiplication With Fractions

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Fraction Operations Activities

Teach your child all about fractions operations with amazing educational resources for children. These online fractions operations learning resources break down the topic into smaller parts for better conceptual understanding and grasp. Get started now to make fractions operations practice a smooth, easy and fun process for your child!

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Add and Subtract Fractions

Estimate the Sum using Number Line Game

Enjoy the marvel of mathematics by exploring how to estimate the sum using a number line.

Estimate the Sum of Two Fractions Game

Have your own math-themed party by learning how to estimate the sum of two fractions.

Estimate the Sum Using Models Worksheet

Put your skills to the test by practicing to estimate the sum using models.

Estimating the Sum Using Models Worksheet

Combine math learning with adventure by estimating the sum using models.

Add and Subtract mixed numbers

Choose the Sum of the Mixed Number and the Fraction Game

Use your math skills to choose the correct sum of the mixed number and the fraction.

Add the Mixed Number and the Fraction on a Number Line Game

Kids must add the mixed number and the fraction on a number line to practice fractions.

Add Mixed Numbers Without Models Worksheet

Put your skills to the test by practicing to add mixed numbers without models.

Add Mixed Numbers Using Models Worksheet

Reveal the secrets of math wizardry by practicing to add mixed numbers using models.

Multiply Fractions

Choose the Correct Addition Sentence Related to the Fraction Game

Kids must choose the correct addition sentence related to the fraction.

Choose the Multiplication Expression Representing Fraction Models Game

Kids must choose the multiplication expression representing fraction models.

Represent Fractions As Sum of Unit Fractions Worksheet

This worksheet will help you represent fractions as the sum of unit fractions.

Express Fractions as Sum of Unit Fractions Worksheet

Combine math learning with adventure by expressing fractions as the sum of unit fractions.

All Fractions Operations Resources

Fill in Colors in the Model to Show the sum Game

Practice fractions by filling in colors in the model to show the sum.

Subtract Fractions using Area Models Worksheet

This downloadable worksheet is designed to subtract fractions using area models.

Find the Sum WIth Help of Fraction Models Game

Practice to find the sum with the help of fraction models in this game.

Subtract Like Fractions using Area Models Worksheet

Sharpen your math skills by subtracting like fractions using area models.

Add Like Fractions using Number Lines Game

Use your fraction skills to add like fractions using number lines.

Find the Sum of the Mixed Numbers with the AId of Models Game

Use your fraction skills to find the sum of mixed numbers with the aid of models.

Subtract Fractions Using Model Worksheet

Dive into this fun-filled printable worksheet by practicing to subtract fractions using models.

Add Mixed Numbers Using Visual Models Worksheet

Help your child revise fractions by solving to add mixed numbers using visual models.

Complete the Multiplication Expression Using the Fraction Models Game

Use fraction models to complete the multiplication expression.

Add Like Fractions Game

Shine bright in the math world by learning how to add like fractions.

Sum of Unit Fractions as Multiplication Expression Worksheet

In this worksheet, kids will identify the sum of unit fractions as multiplication expression.

Subtracting Fractions Using Visual Model Worksheet

Make math practice a joyride by subtracting fractions using visual models.

Add the Fraction to the Mixed Number Game

Use your fraction skills to add the fraction to the mixed number.

Represent Unit Fraction Multiplication on a Number Line Game

Dive deep into the world of math by representing unit fraction multiplication on a number line.

Find the Repeated Sum of a Mixed Number Worksheet

Boost your ability to find the repeated sum of a mixed number by printing this playful worksheet.

Repeated Addition of Fractions as Multiplication Worksheet

Practice repeated addition of fractions as multiplication in this worksheet.

Fill in the Blanks to Complete Fraction Addition Game

Fill in the blanks to complete fraction addition.

FIll in the Blank to add the Fraction and the Mixed Number Game

FIll in the blanks to add the fraction and the mixed number.

Subtract Fractions from 1 Worksheet

Solidify your math skills by practicing to subtract fractions from 1.

Add the Given Mixed Number Repeatedly Worksheet

Pack your math practice time with fun by adding the given mixed number repeatedly.

Multiply Unit Fraction by a Whole Number on a Number Line Game

Kids must multiply unit fractions by a whole number on a number line.

Fill in the Models to FInd the Sum Game

Kids must fill in the models to fInd the sum.

Multiplication of Fractions as Repeated Addition Worksheet

Become a mathematician by practicing multiplication of fractions as repeated addition.

Subtract Fractions from the Wholes using Models Worksheet

Print this worksheet to subtract fractions from the wholes using models like a math legend!

Find the Sum of Mixed Numbers Game

Apply your knowledge of fractions to find the sum of mixed numbers.

Multiply Unit Fraction by Whole Number Game

Enjoy the marvel of math-multiverse by exploring how to multiply unit fractions by whole numbers.

Mixed Numbers Addition Worksheet

Make math practice a joyride by solving problems on mixed numbers addition.

Fraction Multiplication as Sum of Unit Fractions Worksheet

Print this worksheet to practice fraction multiplication as the sum of unit fractions.

Use Fractions Models to FInd the Sum Game

Add more arrows to your child’s math quiver by using fraction models to find the sum.

Complete the Addition of Mixed Numbers Game

Dive deep into the world of fractions by completing the addition of mixed numbers.

Subtract Fractions from the Whole Numbers Worksheet

Focus on core math skills with this fun worksheet by subtracting fractions from whole numbers.

Addition of Mixed Numbers Worksheet

Put your skills to the test by practicing the addition of mixed numbers.

Fill in and Find the Product of Unit Fraction and the Whole Number Game

Kids must fill in and find the product of unit fractions and whole numbers.

Add Fractions WIth the Aid of a Number LIne Game

Ask your little one to add fractions with the aid of a number line to play this game.

Represent a Fraction As Sum of Another Fraction Worksheet

Reveal the secrets of math wizardry by representing a fraction as the sum of another fraction.

Adding Fractions Using Models Worksheet

Learn fractions at the speed of lightning by practicing to add fractions using models.

Cross Out to FInd the Difference Between a Mixed Number and a Fraction Game

Cross out the shaded part to fInd the difference between a mixed number and a fraction.

Select the Multiplication Expression Represented by Fraction Models Game

Select the multiplication expression represented by fraction models correctly.

Adding Mixed Numbers using Models Worksheet

Solidify your math skills by practicing to add mixed numbers using models.

Express a Fraction as Repeated Sum Worksheet

Dive into this fun-filled printable worksheet by practicing to express a fraction as repeated sum.

Your one stop solution for all grade learning needs.

Fraction Worksheets

Conversion :: Addition :: Subtraction :: Multiplication :: Division

Conversions

Fractions - addition, fractions - subtraction, fractions - multiplication, fractions - division.

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4.9: Solving Equations with Fractions

• Last updated
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• Page ID 24084

• David Arnold
• College of the Redwoods

Undoing Subtraction

We can still add the same amount to both sides of an equation without changing the solution.

Solve for x : \(x - \frac{5}{6} = \frac{1}{3}\).

To “undo” subtracting 5/6, add 5/6 to both sides of the equation and simplify.

\[ \begin{aligned} x - \frac{5}{6} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Original equation.}} \\ x - \frac{5}{6} + \frac{5}{6} = \frac{1}{3} + \frac{5}{6} ~ & \textcolor{red}{ \text{ Add } \frac{5}{6} \text{ to both sides.}} \\ x = \frac{1 \cdot 2}{3 \cdot 2} + \frac{5}{6} ~ & \textcolor{red}{ \text{ Equivalent fractions, LCD = 6.}} \\ x = \frac{2}{6} + \frac{5}{6} ~ & \textcolor{red}{ \text{ Simplify.}} \\ x = \frac{7}{6} ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber \]

It is perfectly acceptable to leave your answer as an improper fraction. If you desire, or if you are instructed to do so, you can change your answer to a mixed fraction (7 divided by 6 is 1 with a remainder of 1). That is \(x = 1 \frac{1}{6}\).

Checking the Solution

Substitute 7/6 for x in the original equation and simplify.

\[ \begin{aligned} x - \frac{5}{6} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Original equation.}} \\ \frac{7}{6} - \frac{5}{6} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Substitute 7/6 for } x.} \\ \frac{2}{6} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Subtract.}} \\ \frac{1}{3} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Reduce.}} \end{aligned}\nonumber \]

Because the last statement is true, we conclude that 7/6 is a solution of the equation x − 5/6 = 1/3.

Undoing Addition

You can still subtract the same amount from both sides of an equation without changing the solution.

Solve for x : \(x + \frac{2}{3} = - \frac{3}{5}\).

To “undo” adding 2/3, subtract 2/3 from both sides of the equation and simplify.

\[ \begin{aligned} x + \frac{2}{3} = - \frac{3}{5} ~ & \textcolor{red}{ \text{ Original equation.}} \\ x + \frac{2}{3} - \frac{2}{3} = - \frac{3}{5} - \frac{2}{3} ~ & \textcolor{red}{ \text{ Subtract } \frac{2}{3} \text{ from both sides.}} \\ x = - \frac{3 \cdot 3}{5 \cdot 3} - \frac{2 \cdot 5}{3 \cdot 5} ~ & \textcolor{red}{ \text{ Equivalent fractions, LCD = 15.}} \\ x = - \frac{9}{15} - \frac{10}{15} ~ & \textcolor{red}{ \text{ Simplify.}} \\ x = - \frac{19}{15} ~ & \textcolor{red}{ \text{ Subtract.}} \end{aligned}\nonumber \]

Readers are encouraged to check this solution in the original equation.

Solve for x : \(x + \frac{3}{4} = - \frac{1}{2}\)

Undoing Multiplication

We “undo” multiplication by dividing. For example, to solve the equation 2 x = 6, we would divide both sides of the equation by 2. In similar fashion, we could divide both sides of the equation

\[ \frac{3}{5} x = \frac{4}{10}\nonumber \]

by 3/5. However, it is more efficient to take advantage of reciprocals. For convenience, we remind readers of the Multiplicative Inverse Property .

Multiplicative Inverse Property

Let a / b be any fraction. The number b / a is called the multiplicative inverse or reciprocal of a / b . The product of reciprocals is 1.

\[ \frac{a}{b} \cdot \frac{b}{a} = 1.\nonumber \]

Let’s put our knowledge of reciprocals to work.

Solve for x : \(\frac{3}{5}x = \frac{4}{10}\).

To “undo” multiplying by 3/5, multiply both sides by the reciprocal 5/3 and simplify.

\[ \begin{aligned} \frac{3}{5} x = \frac{4}{10} ~ & \textcolor{red}{ \text{ Original equation.}} \\ \frac{5}{3} \left( \frac{3}{5} x \right) = \frac{5}{3} \left( \frac{4}{10} \right) & ~ \textcolor{red}{ \text{ Multiply both sides by 5/3.}} \\ \left( \frac{5}{3} \cdot \frac{3}{5} \right) x = \frac{20}{30} ~ & \textcolor{red}{ \begin{array}{l} \text{ On the left, use the associative property to regroup.} \\ \text{ On the right, multiply.} \end{array}} \\ 1x = \frac{2}{3} ~ & \textcolor{red}{ \begin{array}{l} \text{ On the left, } \frac{5}{3} \cdot \frac{3}{5} = 1. \\ \text{ On the right, reduce: } \frac{20}{30} = \frac{2}{3}. \end{array}} \\ x = \frac{2}{3} ~ & \textcolor{red}{ \text{ On the left, } 1x = x.} \end{aligned}\nonumber \]

Substitute 2/3 for x in the original equation and simplify.

\[ \begin{aligned} \frac{3}{5} x = \frac{4}{10} ~ & \textcolor{red}{ \text{ Original equation.}} \\ \frac{3}{5} \left( \frac{2}{3} \right) = \frac{4}{10} ~ & \textcolor{red}{ \text{ Substitute 2/3 for }x.} \\ \frac{6}{15} = \frac{4}{10} ~ & \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \\ \frac{2}{5} = \frac{2}{5} ~ & \textcolor{red}{ \text{ Reduce both sides to lowest terms.}} \end{aligned}\nonumber \]

Because this last statement is true, we conclude that 2/3 is a solution of the equation (3/5) x = 4/10.

Solve for y : \( \frac{2}{3} y = \frac{4}{5}\)

Solve for x : \(- \frac{8}{9} x = \frac{5}{18}\).

To “undo” multiplying by −8/9, multiply both sides by the reciprocal −9/8 and simplify.

\[ \begin{aligned} - \frac{8}{9} x = \frac{5}{18} ~ & \textcolor{red}{ \text{ Original equation.}} \\ - \frac{9}{8} \left( - \frac{8}{9} x \right) = - \frac{9}{8} \left( \frac{5}{18} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by } -9/8.} \\ \left[ - \frac{9}{8} \cdot \left( - \frac{8}{9} \right) \right] x = - \frac{3 \cdot 3}{2 \cdot 2 \cdot 2} \cdot \frac{5}{2 \cdot 3 \cdot 3} ~ & \textcolor{red}{ \begin{array}{l} \text{ On the left, use the associative property to regroup.} \\ \text{ On the right, prime factor.} \end{array}} 1x = \frac{ \cancel{3} \cdot \cancel{3}}{2 \cdot 2 \cdot 2} \cdot \frac{5}{2 \cdot \cancel{3} \cdot \cancel{3}} ~ & \textcolor{red}{ \begin{array}{l} \text{ On the left, } - \frac{9}{8} \cdot \left( - \frac{8}{9} \right) = 1. \\ \text{ On the right, cancel common factors.} \end{array}} \\ x = - \frac{5}{16} ~ & \textcolor{red}{ \text{ On the left, } 1x = x. \text{ Multiply on the right.}} \end{aligned}\nonumber \]

Solve for z: \(− \frac{2}{7} z = \frac{4}{21}\)

Clearing Fractions from the Equation

Although the technique demonstrated in the previous examples is a solid mathematical technique, working with fractions in an equation is not always the most efficient use of your time.

To clear all fractions from an equation, multiply both sides of the equation by the least common denominator of the fractions that appear in the equation.

Let’s put this idea to work.

In Example 1, we were asked to solve the following equation for x :

\[x - \frac{5}{6} = \frac{1}{3}.\nonumber \]

Take a moment to review the solution technique in Example 1. We will now solve this equation by first clearing all fractions from the equation.

Multiply both sides of the equation by the least common denominator for the fractions appearing in the equation.

\[ \begin{aligned} x - \frac{5}{6}= \frac{1}{3} ~ & \textcolor{red}{ \text{ Original equation.}} \\ 6 \left( x - \frac{5}{6} \right) = 6 \left( \frac{1}{3} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 6.}} \\ 6x - 6 \left( \frac{5}{6} \right) = 6 \left( \frac{1}{3} \right) ~ & \textcolor{red}{ \text{ Distribute the 6.}} \\ 6x-5 = 2 ~ & \textcolor{red}{ \text{ On each side, multiply first.}} \\ ~ & \textcolor{red}{6 \left( \frac{5}{6} \right) = 5 \text{ and } 6 \left( \frac{1}{3} \right) = 2.} \end{aligned}\nonumber \]

Note that the equation is now entirely clear of fractions, making it a much simpler equation to solve.

\[ \begin{aligned} 6x - 5 + 5 = 2 + 5 ~ & \textcolor{red}{ \text{ Add 5 to both sides.}} \\ 6x = 7 ~ & \textcolor{red}{ \text{ Simplify both sides.}} \\ \frac{6x}{6} = \frac{7}{6} ~ & \textcolor{red}{ \text{ Divide both sides by 6.}} \\ x = \frac{7}{6} ~ & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]

Note that this is the same solution found in Example 1.

Solve for t : \(t - \frac{2}{7} = - \frac{1}{4}\)

In Example 4, we were asked to solve the following equation for x .

\[- \frac{8}{9}x = \frac{5}{18}\nonumber \]

Take a moment to review the solution in Example 4. We will now solve this equation by first clearing all fractions from the equation.

Multiply both sides of the equation by the least common denominator for the fractions that appear in the equation.

\[ \begin{aligned} - \frac{8}{9} x = \frac{5}{18} ~ & \textcolor{red}{ \text{ Original equation.}} \\ 18 \left( - \frac{8}{9} x \right) = 18 \left( \frac{5}{18} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 18.}} \\ -16x=5 ~ & \textcolor{red}{ \text{ On each side, cancel and multiply.}} \\ ~ & \textcolor{red}{ 18 \left( - \frac{8}{9} \right) = -16 \text{ and } 18 \left( \frac{5}{18} \right) = 5.} \end{aligned}\nonumber \]

Note that the equation is now entirely free of fractions. Continuing,

\[ \begin{aligned} \frac{-16x}{-16} = \frac{5}{-16} ~ & \textcolor{red}{ \text{ Divide both sides by } -16.} \\ x = - \frac{5}{16} ~ & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]

Note that this is the same as the solution found in Example 4.

Solve for u :

\[ - \frac{7}{9} u = \frac{14}{27}\nonumber \]

Solve for x : \(\frac{2}{3}x + \frac{3}{4} = \frac{1}{2}\).

\[ \begin{aligned} \frac{2}{3} x + \frac{3}{4} = \frac{1}{2} ~ & \textcolor{red}{ \text{ Original equation.}} \\ 12 \left( \frac{2}{3} x + \frac{3}{4} = \right) = 12 \left( \frac{1}{2} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 12.}} \\ 12 \left( \frac{2}{3}x \right) + 12 \left( \frac{3}{4} \right) = 12 \left( \frac{1}{2} \right) ~ & \textcolor{red}{ \text{ On the left, distribute 12.}} \\ 8x + 9 = 6 ~ & \textcolor{red}{ \text{ Multiply: } 12 \left( \frac{2}{3} x \right) = 8x, ~ 12 \left( \frac{3}{4} \right) = 9,} \\ ~ & \textcolor{red}{ \text{ and } 12 \left( \frac{1}{2} \right) = 6.} \end{aligned}\nonumber \]

Note that the equation is now entirely free of fractions. We need to isolate the terms containing x on one side of the equation.

\[ \begin{aligned} 8x + 9 - 9 = 6 - 9 ~ & \textcolor{red}{ \text{ Subtract 9 from both sides.}} \\ 8x = - 3 ~ & \textcolor{red}{ \text{ Simplify both sides.}} \\ \frac{8x}{8} = \frac{-3}{8} ~ & \textcolor{red}{ \text{ Divide both sides by 8.}} \\ x = - \frac{3}{8} ~ & \textcolor{red}{ \text{ Simplify both sides.}} \end{aligned}\nonumber \]

Solve for r : \(\frac{3}{4} r + \frac{2}{3} = \frac{1}{2}\)

Solve for x : \( \frac{2}{3} - \frac{3x}{4} = \frac{x}{2} - \frac{1}{8}.\)

Multiply both sides of the equation by the least common denominator for the fractions in the equation.

\[ \begin{aligned} \frac{2}{3} - \frac{3x}{4} = \frac{x}{2} - \frac{1}{8} ~ & \textcolor{red}{ \text{ Original equation.}} \\ 24 \left( \frac{2}{3} - \frac{3x}{4} \right) = 24 \left( \frac{x}{2} - \frac{1}{8} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 24.}} \\ 24 \left( \frac{2}{3} \right) - 24 \left( \frac{3x}{4} \right) = 24 \left( \frac{x}{2} \right) - 24 \left( \frac{1}{8} \right) ~ & \textcolor{red}{ \text{ On both sides, distribute 24.}} \\ 16 - 18x = 12x - 3 ~ & \textcolor{red}{ \text{ Left: } 24 \left( \frac{2}{3} \right) = 16, ~ 24 \left( \frac{3x}{4} \right) = 18x.} \\ ~ & \textcolor{red}{ \text{ Right: } 24 \left( \frac{x}{2} \right) = 12x, ~ 24 \left( \frac{1}{8} \right) = 3.} \end{aligned}\nonumber \]

\[ \begin{aligned} 16 - 18x - 12x = 12x - 3 - 12x ~ & \textcolor{red}{ \text{ Subtract } 12x \text{ from both sides.}} \\ 16 - 30x = -3 ~ & \textcolor{red}{ \begin{aligned} \text{ Left: } -18x - 12x = -30x. \\ \text{ Right: } 12x - 12x = 0. \end{aligned}} \\ 16 - 30x - 16 = -3 - 16 ~ & \textcolor{red}{ \text{ Subtract 16 from both sides.}} \\ -30x = -19 ~ & \textcolor{red}{ \begin{aligned} \text{ Left: } 16-16=0. \\ \text{ Right: } -3 - 16 = -19. \end{aligned}} \\ \frac{-30x}{-30} = \frac{-19}{-30} ~ & \textcolor{red}{ \text{ Divide both sides by } -30.} \\ x = \frac{19}{30} ~ & \textcolor{red}{ \text{ Simplify both sides.}} \end{aligned}\nonumber \]

Solve for s : \( \frac{3}{2} - \frac{2s}{5} = \frac{s}{3} - \frac{1}{5}\).

Add texts here. Do not delete this text first.

Applications

Let’s look at some applications that involve equations containing fractions. For convenience, we repeat the Requirements for Word Problem Solutions .

Requirements for Word Problem Solutions

• Statements such as “Let P represent the perimeter of the rectangle.”
• Labeling unknown values with variables in a table.
• Labeling unknown quantities in a sketch or diagram.
• Set up an Equation . Every solution to a word problem must include a carefully crafted equation that accurately describes the constraints in the problem statement.
• Solve the Equation . You must always solve the equation set up in the previous step.
• Answer the Question . This step is easily overlooked. For example, the problem might ask for Jane’s age, but your equation’s solution gives the age of Jane’s sister Liz. Make sure you answer the original question asked in the problem. Your solution should be written in a sentence with appropriate units. 5. Look Back. It is important to note that this step does not imply that you should simply check your solution in your equation. After all, it’s possible that your equation incorrectly models the problem’s situation, so you could have a valid solution to an incorrect equation. The important question is: “Does your answer make sense based on the words in the original problem statement.”

In the third quarter of a basketball game, announcers informed the crowd that attendance for the game was 12,250. If this is two-thirds of the capacity, find the full seating capacity for the basketball arena.

We follow the Requirements for Word Problem Solutions .

1. Set up a Variable Dictionary . Let F represent the full seating capacity. Note: It is much better to use a variable that “sounds like” the quantity that it represents. In this case, letting F represent the full seating capacity is much more descriptive than using x to represent the full seating capacity.

2. Set up an Equation . Two-thirds of the full seating capacity is 12,250.

\[ \begin{aligned} \colorbox{cyan}{Two-thirds} & \text{ of } & \colorbox{cyan}{Full Seating Capacity} & \text{ is } & 12,250 \\ \frac{2}{3} & \cdot & F & = & 12,250 \end{aligned}\nonumber \]

Hence, the equation is

\[ \frac{2}{3} F = 12250.\nonumber \]

3. Solve the Equation . Multiply both sides by 3 to clear fractions, then solve.

\[ \begin{aligned} \frac{2}{3} F = 12250 ~ & \textcolor{red}{ \text{ Original equation.}} \\ 3 \left( \frac{2}{3} F \right) = 3(12250) ~ & \textcolor{red}{ \text{ Multiply both sides by 3.}} \\ 2F = 36750 ~ & \textcolor{red}{ \text{ Simplify both sides.}} \\ \frac{2F}{2} = \frac{36750}{2} ~ & \textcolor{red}{ \text{ Divide both sides by 2.}} \\ F = 18375 ~ & \textcolor{red}{ \text{ Simplify both sides.}} \end{aligned}\nonumber \]

4. Answer the Question . The full seating capacity is 18,375.

5. Look Back . The words of the problem state that 2/3 of the seating capacity is 12,250. Let’s take two-thirds of our answer and see what we get.

\[ \begin{aligned} \frac{2}{3} \cdot 18375 & = \frac{2}{3} \cdot \frac{18375}{1} \\ & = \frac{2}{3} \cdot \frac{3 \cdot 6125}{1} \\ & = \frac{2}{ \cancel{3}} \cdot \frac{ \cancel{3} \cdot 6125}{1} \\ & = 12250 \end{aligned}\nonumber \]

This is the correct attendance, so our solution is correct.

Attendance for the Celtics game was 9,510. If this is 3/4 of capacity, what is the capacity of the Celtics’ arena?

The area of a triangle is 20 square inches. If the length of the base is \(2 \frac{1}{2}\) inches, find the height (altitude) of the triangle.

1. Set up a Variable Dictionary . Our variable dictionary will take the form of a well labeled diagram.

2. Set up an Equation . The area A of a triangle with base b and height h is

\[A = \frac{1}{2} bh.\nonumber \]

Substitute A = 20 and b = \(2 \frac{1}{2}\).

\[20 = \frac{1}{2} \left( 2 \frac{1}{2} \right) h.\nonumber \]

3. Solve the Equation . Change the mixed fraction to an improper fraction, then simplify.

\[ \begin{aligned} 20 = \frac{1}{2} \left( 2 \frac{1}{2} \right) h ~ & \textcolor{red}{ \text{ Original equation.}} \\ 20 = \frac{1}{2} \left( \frac{5}{2} \right) h ~ & \textcolor{red}{ \text{ Mixed to improper: } 2 \frac{1}{2} = \frac{5}{2}.} \\ 20 = \left( \frac{1}{2} \cdot \frac{5}{2} \right) h ~ & \textcolor{red}{ \text{ Associative property.}} \\ 20 = \frac{5}{4} h ~ & \textcolor{red}{ \text{ Multiply: } \frac{1}{2} \cdot \frac{5}{2} = \frac{5}{4}.} \end{aligned}\nonumber \]

Now, multiply both sides by 4/5 and solve.

\[ \begin{aligned} \frac{4}{5} (20) = \frac{4}{5} \left( \frac{5}{4} h \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 4/5.}} \\ 16 = h ~ & \textcolor{red}{ \text{ Simplify: } \frac{4}{5} (20) = 16} \\ ~ & \textcolor{red}{ \text{ and } \frac{4}{5} \cdot \frac{5}{4} = 1.} \end{aligned}\nonumber \]

4. Answer the Question . The height of the triangle is 16 inches.

5. Look Back . If the height is 16 inches and the base is \(2 \frac{1}{2}\) inches, then the area is

\[ \begin{aligned} A & = \frac{1}{2} \left( 2 \frac{1}{2} \right) (16) \\ & = \frac{1}{2} \cdot \frac{5}{2} \cdot \frac{16}{1} \\ & = \frac{5 \cdot 16}{2 \cdot 2} \\ & = \frac{(5) \cdot (2 \cdot 2 \cdot 2 \cdot 2)}{(2) \cdot (2)} \\ & = \frac{5 \cdot \cancel{2} \cdot \cancel{2} \cdot 2 \cot 2}{ \cancel{2} \cdot \cancel{2}} & = 20 \end{aligned}\nonumber \]

This is the correct area (20 square inches), so our solution is correct.

The area of a triangle is 161 square feet. If the base of the triangle measures \(40 \frac{1}{4}\) feet, find the height of the triangle.

1. Is 1/4 a solution of the equation \(x + \frac{5}{8} = \frac{5}{8}\)?

2. Is 1/4 a solution of the equation \(x + \frac{1}{3} = \frac{5}{12}\)?

3. Is −8/15 a solution of the equation \(\frac{1}{4} x = − \frac{1}{15}\)?

4. Is −18/7 a solution of the equation \(− \frac{3}{8} x = \frac{25}{28}\)?

5. Is 1/2 a solution of the equation \(x + \frac{4}{9} = \frac{17}{18}\)?

6. Is 1/3 a solution of the equation \(x + \frac{3}{4} = \frac{13}{12}\)?

7. Is 3/8 a solution of the equation \(x − \frac{5}{9} = − \frac{13}{72}\)?

8. Is 1/2 a solution of the equation \(x − \frac{3}{5} = − \frac{1}{10}\)?

9. Is 2/7 a solution of the equation \(x − \frac{4}{9} = − \frac{8}{63}\)?

10. Is 1/9 a solution of the equation \(x − \frac{4}{7} = − \frac{31}{63}\)?

11. Is 8/5 a solution of the equation \( \frac{11}{14}x = \frac{44}{35}\)?

12. Is 16/9 a solution of the equation \(\frac{13}{18} x = \frac{104}{81}\)?

In Exercises 13-24, solve the equation and simplify your answer.

13. \(2x − 3=6x + 7\)

14. \(9x − 8 = −9x − 3\)

15. \(−7x +4=3x\)

16. \(6x +9= −6x\)

17. \(−2x = 9x − 4\)

18. \(−6x = −9x + 8\)

19. \(−8x = 7x − 7\)

20. \(−6x = 5x + 4\)

21. \(−7x +8=2x\)

22. \(−x − 7=3x\)

23. \(−9x +4=4x − 6\)

24. \(−2x +4= x − 7\)

In Exercises 25-48, solve the equation and simplify your answer.

25. \(x + \frac{3}{2 = \frac{1}{2}\)

26. \(x − \frac{3}{4} = \frac{1}{4}\)

27. \(− \frac{9}{5} x = \frac{1}{2}\)

28. \(\frac{7}{3} x = − \frac{7}{2}\)

29. \(\frac{3}{8} x = \frac{8}{7}\)

30. \(− \frac{1}{9} x = − \frac{3}{5}\)

31. \(\frac{2}{5} x = − \frac{1}{6}\)

32. \(\frac{1}{6} x = \frac{2}{9}\)

33. \(− \frac{3}{2} x = \frac{8}{7}\)

34. \(− \frac{3}{2} x = − \frac{7}{5}\)

35. \(x + \frac{3}{4} = \frac{5}{9}\)

36. \(x − \frac{1}{9} = − \frac{3}{2}\)

37. \(x − \frac{4}{7} = \frac{7}{8}\)

38. \(x + \frac{4}{9} = − \frac{3}{4}\)

39. \(x + \frac{8}{9} = \frac{2}{3}\)

40. \(x − \frac{5}{6} = \frac{1}{4}\)

41. \(x + \frac{5}{2} = − \frac{9}{8}\)

42. \(x + \frac{1}{2} = \frac{5}{3}\)

43. \(− \frac{8}{5} x = \frac{7}{9}\)

44. \(− \frac{3}{2} x = − \frac{5}{9}\)

45. \(x − \frac{1}{4} = − \frac{1}{8}\)

46. \(x − \frac{9}{2} = − \frac{7}{2}\)

47. \(− \frac{1}{4} x = \frac{1}{2}\)

48. \(− \frac{8}{9} x = − \frac{8}{3}\)

In Exercises 49-72, solve the equation and simplify your answer.

49. \(− \frac{7}{3} x − \frac{2}{3} = \frac{3}{4} x + \frac{2}{3}\)

50. \(\frac{1}{2} x − \frac{1}{2} = \frac{3}{2} x + \frac{3}{4}\)

51. \(− \frac{7}{2} x − \frac{5}{4} = \frac{4}{5}\)

52. \(− \frac{7}{6} x + \frac{5}{6} = − \frac{8}{9}\)

53. \(− \frac{9}{7} x + \frac{9}{2} = − \frac{5}{2}\)

54. \(\frac{5}{9} x − \frac{7}{2} = \frac{1}{4}\)

55. \(\frac{1}{4} x − \frac{4}{3} = − \frac{2}{3}\)

56. \(\frac{8}{7} x + \frac{3}{7} = \frac{5}{3}\)

57. \(\frac{5}{3} x + \frac{3}{2} = − \frac{1}{4}\)

58. \(\frac{1}{2} x − \frac{8}{3} = − \frac{2}{5}\)

59. \(− \frac{1}{3} x + \frac{4}{5} = − \frac{9}{5} x − \frac{5}{6}\)

60. \(− \frac{2}{9} x − \frac{3}{5} = \frac{4}{5} x − \frac{3}{2}\)

61. \(− \frac{4}{9} x − \frac{8}{9} = \frac{1}{2} x − \frac{1}{2}\)

62. \(− \frac{5}{4} x − \frac{5}{3} = \frac{8}{7} x + \frac{7}{3}\)

63. \(\frac{1}{2} x − \frac{1}{8} = − \frac{1}{8} x + \frac{5}{7}\)

64. \(− \frac{3}{2} x + \frac{8}{3} = \frac{7}{9} x − \frac{1}{2}\)

65. \(− \frac{3}{7} x − \frac{1}{3} = − \frac{1}{9}\)

66. \(\frac{2}{3} x + \frac{2}{9} = − \frac{9}{5}\)

67. \(− \frac{3}{4} x + \frac{2}{7} = \frac{8}{7} x − \frac{1}{3}\)

68. \(\frac{1}{2} x + \frac{1}{3} = − \frac{5}{2} x − \frac{1}{4}\)

69. \(− \frac{3}{4} x − \frac{2}{3} = − \frac{2}{3} x − \frac{1}{2}\)

70. \(\frac{1}{3} x − \frac{5}{7} = \frac{3}{2} x + \frac{4}{3}\)

71. \(− \frac{5}{2} x + \frac{9}{5} = \frac{5}{8}\)

72. \(\frac{9}{4} x + \frac{4}{3} = − \frac{1}{6}\)

73. At a local soccer game, announcers informed the crowd that attendance for the game was 4,302. If this is 2/9 of the capacity, find the full seating capacity for the soccer stadium.

74. At a local basketball game, announcers informed the crowd that attendance for the game was 5,394. If this is 2/7 of the capacity, find the full seating capacity for the basketball stadium.

75. The area of a triangle is 51 square inches. If the length of the base is \(8 \frac{1}{2}\) inches, find the height (altitude) of the triangle.

76. The area of a triangle is 20 square inches. If the length of the base is \(2 \frac{1}{2}\) inches, find the height (altitude) of the triangle.

77. The area of a triangle is 18 square inches. If the length of the base is \(4 \frac{1}{2}\) inches, find the height (altitude) of the triangle.

78. The area of a triangle is 44 square inches. If the length of the base is \(5 \frac{1}{2}\) inches, find the height (altitude) of the triangle.

79. At a local hockey game, announcers informed the crowd that attendance for the game was 4,536. If this is 2/11 of the capacity, find the full seating capacity for the hockey stadium.

80. At a local soccer game, announcers informed the crowd that attendance for the game was 6,970. If this is 2/7 of the capacity, find the full seating capacity for the soccer stadium.

81. Pirates . About one-third of the world’s pirate attacks in 2008 occurred off the Somali coast. If there were 111 pirate attacks off the Somali coast, estimate the number of pirate attacks worldwide in 2008.

82. Nuclear arsenal . The U.S. and Russia agreed to cut nuclear arsenals of long-range nuclear weapons by about a third, down to 1, 550. How many long-range nuclear weapons are there now? Associated Press-Times-Standard 04/04/10 Nuclear heartland anxious about missile cuts.

83. Seed vault . The Svalbard Global Seed Vault has amassed half a million seed samples, and now houses at least one-third of the world’s crop seeds. Estimate the total number of world’s crop seeds. Associated Press-Times-Standard 03/15/10 Norway doomsday seed vault hits half-million mark.

84. Freight train . The three and one-half mile long Union Pacific train is about 2 1 2 times the length of a typical freight train. How long is a typical freight train? Associated Press-Times-Standard 01/13/10 Unusally long train raises safety concerns.

13. \(− \frac{5}{2}\)

15. \(\frac{2}{5}\)

17. \(\frac{4}{11}\)

19. \(\frac{7}{15}\)

21. \(\frac{8}{9}\)

23. \(\frac{10}{13}\)

25. \(−1\)

27. \(− \frac{5}{18}\)

29. \(\frac{64}{21}\)

31. \(− \frac{5}{12}\)

33. \(− \frac{16}{21}\)

35. \(− \frac{7}{36}\)

37. \(\frac{81}{56}\)

39. \(− \frac{2}{9}\)

41. \(− \frac{29}{8}\)

43. \(− \frac{35}{72}\)

45. \(\frac{1}{8}\)

47. \(−2\)

49. \(− \frac{16}{37}\)

51. \(− \frac{41}{70}\)

53. \(\frac{49}{9}\)

55. \(\frac{8}{3}\)

57. \(− \frac{21}{20}\)

59. \(− \frac{49}{44}\)

61. \(− \frac{7}{17}\)

63. \(\frac{47}{35}\)

65. \(− \frac{14}{27}\)

67. \(\frac{52}{159}\)

69. \(− 2\)

71. \(\frac{47}{100}\)

81. There were about 333 pirate attacks worldwide.

83. 1,500,000

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Fractions: mixed operations

Fraction word problems with the 4 operations.

These word problems involve the 4 basic operations ( addition, subtraction, multiplication and division ) on fractions .  Mixing word problems encourages students to read and think about the questions, rather than simply recognizing a pattern to the solutions.  

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