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Subtracting fractions word problems

Subtracting fractions word problems: 4 real-life examples.

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Adding fractions word problems

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Subtracting Fractions

You might like to read Adding Fractions first.

There are 3 simple steps to subtract fractions

• Step 1. Make sure the bottom numbers (the denominators) are the same
• Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator.
• Step 3. Simplify the fraction (if needed).

Step 1 . The bottom numbers are already the same. Go straight to step 2.

Step 2 . Subtract the top numbers and put the answer over the same denominator:

3 4 − 1 4   =   3 − 1 4   =   2 4

Step 3 . Simplify the fraction:

(If you are unsure of the last step see Equivalent Fractions .)

Step 1 . The bottom numbers are different. See how the slices are different sizes? We need to make them the same before we can continue, because we can't subtract them like this:

To make the bottom numbers the same, multiply the top and bottom of the first fraction ( 1 / 2 ) by 3 like this:

And now our question looks like this:

The bottom numbers (the denominators) are the same, so we can go to step 2.

3 6 − 1 6   =   3 − 1 6   =   2 6

In picture form it looks like this:

With Pen and Paper

And here is how to do it with a pen and paper (press the play button):

Subtracting Mixed Fractions

I have a special page on Adding and Subtracting Mixed Fractions .

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• reduce\:fraction\:\frac{4}{8}
• \frac{1}{2}+\frac{1}{4}+\frac{3}{4}
• \frac{1}{2}\cdot\frac{8}{7}
• \frac{-\frac{1}{5}}{\frac{7}{4}}
• descending\:order\:\frac{1}{2},\:\frac{3}{6},\:\frac{7}{2}
• decimal\:to\:fraction\:0.35
• What is a mixed number?
• A mixed number is a combination of a whole number and a fraction.
• How can I compare two fractions?
• To compare two fractions, first find a common denominator, then compare the numerators.Alternatively, compare the fractions by converting them to decimals.
• How do you add or subtract fractions with different denominators?
• To add or subtract fractions with different denominators, convert the fractions to have a common denominator. Then you can add or subtract the numerators of the fractions, leaving the denominator unchanged.

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Fractions  - Adding and Subtracting Fractions

Fractions  -, adding and subtracting fractions, fractions adding and subtracting fractions.

Fractions: Adding and Subtracting Fractions

Lesson 3: adding and subtracting fractions.

/en/fractions/comparing-and-reducing-fractions/content/

Adding and subtracting fractions

In the previous lessons, you learned that a fraction is part of a whole. Fractions show how much you have of something, like 1/2 of a tank of gas or 1/3 of a cup of water.

In real life, you might need to add or subtract fractions. For example, have you ever walked 1/2 of a mile to work and then walked another 1/2 mile back? Or drained 1/4 of a quart of gas from a gas tank that had 3/4 of a quart in it? You probably didn't think about it at the time, but these are examples of adding and subtracting fractions.

Click through the slideshow to learn how to set up addition and subtraction problems with fractions.

Let's imagine that a cake recipe tells you to add 3/5 of a cup of oil to the batter.

You also need 1/5 of a cup of oil to grease the pan. To see how much oil you'll need total, you can add these fractions together.

When you add fractions, you just add the top numbers, or numerators .

That's because the bottom numbers, or denominators , show how many parts would make a whole.

We don't want to change how many parts make a whole cup ( 5 ). We just want to find out how many parts we need total.

So we only need to add the numerators of our fractions.

We can stack the fractions so the numerators are lined up. This will make it easier to add them.

And that's all we have to do to set up an addition example with fractions. Our fractions are now ready to be added.

We'll do the same thing to set up a subtraction example. Let's say you had 3/4 of a tank of gas when you got to work.

If you use 1/4 of a tank to drive home, how much will you have left? We can subtract these fractions to find out.

Just like when we added, we'll stack our fractions to keep the numerators lined up.

This is because we want to subtract 1 part from 3 parts.

Now that our example is set up, we're ready to subtract!

Try setting up these addition and subtraction problems with fractions. Don't try solving them yet!

You run 4/10 of a mile in the morning. Later, you run for 3/10 of a mile.

You had 7/8 of a stick of butter and used 2/8 of the stick while cooking dinner.

Your gas tank is 2/5 full, and you put in another 2/5 of a tank.

Solving addition problems with fractions

Now that we know how to write addition problems with fractions, let's practice solving a few. If you can add whole numbers , you're ready to add fractions.

Click through the slideshow to learn how to add fractions.

Let's continue with our previous example and add these fractions: 3/5 of cup of oil and 1/5 of a cup of oil.

Remember, when we add fractions, we don't add the denominators.

This is because we're finding how many parts we need total. The numerators show the parts we need, so we'll add 3 and 1 .

3 plus 1 equals 4 . Make sure to line up the 4 with the numbers you just added.

The denominators will stay the same, so we'll write 5 on the bottom of our new fraction.

3/5 plus 1/5 equals 4/5 . So you'll need 4/5 of a cup of oil total to make your cake.

Let's try another example: 7/10 plus 2/10 .

Just like before, we're only going to add the numerators. In this example, the numerators are 7 and 2 .

7 plus 2 equals 9 , so we'll write that to the right of the numerators.

Just like in our earlier example, the denominator stays the same.

So 7/10 plus 2/10 equals 9/10 .

Try solving some of the addition problems below.

Solving subtraction problems with fractions

Subtracting fractions is a lot like regular subtraction. If you can subtract whole numbers , you can subtract fractions too!

Click through the slideshow to learn how to subtract fractions.

Let's use our earlier example and subtract 1/4 of a tank of gas from 3/4 of a tank.

Just like in addition, we're not going to change the denominators.

We don't want to change how many parts make a whole tank of gas. We just want to know how many parts we'll have left.

We'll start by subtracting the numerators. 3 minus 1 equals 2 , so we'll write 2 to the right of the numerators.

Just like when we added, the denominator of our answer will be the same as the other denominators.

So 3/4 minus 1/4 equals 2/4 . You'll have 2/4 of a tank of gas left when you get home.

Let's try solving another problem: 5/6 minus 3/6 .

We'll start by subtracting the numerators.

5 minus 3 equals 2 . So we'll put a 2 to the right of the numerators.

As usual, the denominator stays the same.

So 5/6 minus 3/6 equals 2/6 .

Try solving some of the subtraction problems below.

After you add or subtract fractions, you may sometimes have a fraction that can be reduced to a simpler fraction. As you learned in Comparing and Reducing Fractions , it's always best to reduce a fraction to its simplest form when you can. For example, 1/4 plus 1/4 equals 2/4 . Because 2 and 4 can both be divided 2 , we can reduce 2/4 to 1/2 .

Adding fractions with different denominators

On the last page, we learned how to add fractions that have the same denominator, like 1/4 and 3/4 . But what if you needed to add fractions with different denominators? For example, our cake recipe might say to blend 1/4 cup of milk in slowly and then dump in another 1/3 of a cup.

In Comparing and Reducing Fractions , we compared fractions with a different bottom number, or denominator. We had to change the fractions so their denominators were the same. To do that, we found the lowest common denominator , or LCD .

We can only add or subtract fractions if they have the same denominators. So we'll need to find the lowest common denominator before we add or subtract these fractions. Once the fractions have the same denominator, we can add or subtract as usual.

Click through the slideshow to learn how to add fractions with different denominators.

Let's add 1/4 and 1/3 .

Before we can add these fractions, we'll need to change them so they have the same denominator .

To do that, we'll have to find the LCD , or lowest common denominator, of 4 and 3 .

It looks like 12 is the smallest number that can be divided by both 3 and 4, so 12 is our LCD .

Since 12 is the LCD, it will be the new denominator for our fractions.

Now we'll change the numerators of the fractions, just like we changed the denominators.

First, let's look at the fraction on the left: 1/4 .

To change 4 into 12 , we multiplied it by 3 .

Since the denominator was multiplied by 3 , we'll also multiply the numerator by 3 .

1 times 3 equals 3 .

1/4 is equal to 3/12 .

Now let's look at the fraction on the right: 1/3 . We changed its denominator to 12 as well.

Our old denominator was 3 . We multiplied it by 4 to get 12.

We'll also multiply the numerator by 4 . 1 times 4 equals 4 .

So 1/3 is equal to 4/12 .

Now that our fractions have the same denominator, we can add them like we normally do.

3 plus 4 equals 7 . As usual, the denominator stays the same. So 3/12 plus 4/12 equals 7/12 .

Try solving the addition problems below.

Subtracting fractions with different denominators

We just saw that fractions can only be added when they have the same denominator. The same thing is true when we're subtracting fractions. Before we can subtract, we'll have to change our fractions so they have the same denominator.

Click through the slideshow to learn how to subtract fractions with different denominators.

Let's try subtracting 1/3 from 3/5 .

First, we'll change the denominators of both fractions to be the same by finding the lowest common denominator .

It looks like 15 is the smallest number that can be divided evenly by 3 and 5 , so 15 is our LCD.

Now we'll change our first fraction. To change the denominator to 15 , we'll multiply the denominator and the numerator by 3 .

5 times 3 equals 15 . So our fraction is now 9/15 .

Now let's change the second fraction. To change the denominator to 15 , we'll multiply both numbers by 5 to get 5/15 .

Now that our fractions have the same denominator, we can subtract like we normally do.

9 minus 5 equals 4 . As always, the denominator stays the same. So 9/15 minus 5/15 equals 4/15 .

Try solving the subtraction problems below.

Adding and subtracting mixed numbers

Over the last few pages, you've practiced adding and subtracting different kinds of fractions. But some problems will need one extra step. For example, can you add the fractions below?

In Introduction to Fractions , you learned about mixed numbers . A mixed number has both a fraction and a whole number . An example is 2 1/2 , or two-and-a-half . Another way to write this would be 5/2 , or five-halves . These two numbers look different, but they're actually the same.

5/2 is an improper fraction . This just means the top number is larger than the bottom number. Even though improper fractions look strange, you can add and subtract them just like normal fractions. Mixed numbers aren't easy to add, so you'll have to convert them into improper fractions first.

Let's add these two mixed numbers: 2 3/5 and 1 3/5 .

We'll need to convert these mixed numbers to improper fractions. Let's start with 2 3/5 .

As you learned in Lesson 2 , we'll multiply the whole number, 2 , by the bottom number, 5 .

2 times 5 equals 10 .

Now, let's add 10 to the numerator, 3 .

10 + 3 equals 13 .

Just like when you add fractions, the denominator stays the same. Our improper fraction is 13/5 .

Now we'll need to convert our second mixed number: 1 3/5 .

First, we'll multiply the whole number by the denominator. 1 x 5 = 5 .

Next, we'll add 5 to the numerators. 5 + 3 = 8 .

Just like last time, the denominator remains the same. So we've changed 1 3/5 to 8/5 .

Now that we've changed our mixed numbers to improper fractions, we can add like we normally do.

13 plus 8 equals 21 . As usual, the denominator will stay the same. So 13/5 + 8/5 = 21/5 .

Because we started with a mixed number, let's convert this improper fraction back into a mixed number.

As you learned in the previous lesson , divide the top number by the bottom number. 21 divided by 5 equals 4, with a remainder of 1 .

The answer, 4, will become our whole number.

And the remainder , 1, will become the numerator of the fraction.

So 2 3/5 + 1 3/5 = 4 1/5 .

/en/fractions/multiplying-and-dividing-fractions/content/

• Math Article

Subtracting Fractions

Subtracting fractions include the subtraction of two or more fractions with the same or different denominators. Like fractions can be subtracted directly but for unlike fractions we need to make the denominators same first and then subtract them.  In Mathematics, a fraction is a portion of a quantity out of the whole. The whole quantity can be any number, special value or item. We can perform different arithmetic operations on fractions such as addition, subtraction, multiplication and division.

In this article, we will learn what are subtracting fractions and subtracting fractions with like denominators, unlike denominators, and whole numbers. Also, learn the subtraction of mixed fractions with many solved examples.

Introduction to Fractions

A fraction is a numerical value that represents the parts of the whole. A fraction consists of two parts, namely the numerator and the denominator. The upper part of the fraction is called the numerator and the lower part of the fraction is called the denominator. For example, 7/9 is a fraction. Here, 7 is the numerator and 9 is the denominator. Based on the numerator and the denominator, there are different types of fractions . They are:

Proper Fraction : In a proper fraction, the numerator is less than the denominator. Example: ⅗, ⅖, etc.

Improper Fraction : In improper fractions, the numerator is greater than the denominator. Example, 9/7, 11/9, etc.

Mixed Fraction : The mixed fraction is the combination of the proper fraction and a whole number. Example 2 ⅘, 4 ⅔, etc.

Unit Fraction: In unit fraction, the numerator should be equal to 1. For example, ⅓, ¼, ⅕, etc.

Equivalent Fractions : The equivalent fractions are the fractions that represent the same value. If we multiply or divide the numerator and the denominator by the same value, we get the equivalent fractions. Example, 2/4, 4/8, 8/16, etc

Like Fractions : Fractions with the same denominators are called the like fractions. Example: 3/2, 5/2, 7/2, etc

Unlike Fractions : Fractions with different denominators are called unlike fractions. Example: 2/7, 2/9, 3/11, and so on.

What is Meant by Subtracting Fractions?

In Mathematics, subtracting fractions means the process of the subtraction of two fractional values. We have learned to subtract the whole numbers . For example, the subtraction of 3 from 5 results in 2. (i.e. 5-3 = 2). Similarly, we can perform subtraction operations on fractions. Subtracting Fractions include:

Subtracting Fractions with Like Denominators

Subtracting fractions with unlike denominators, subtracting mixed fractions, subtracting fractions with whole numbers.

Now, let us discuss all these subtracting fractions in detail with examples.

Subtracting fractions with like denominators means the subtraction of fractions with the same denominator values. Follow the below steps to subtract the like fractions.

Step 1: Keep the denominator values as it is and subtract the numerator value, which will give the result.

Step 2: If required, simplify the fraction.

Subtract 7/12 from 9/12.

Given: (9/12) – (7/12)

Here, the denominator values are the same and keep the value as it is.

Now, subtract the numerator values

(9/12) – (7/12) = (9-7)/12

(9/12) – (7/12) = 2/12

Simplify the fraction, and we get

(9/12) – (7/12) = 1/6.

Therefore, (9/12) – (7/12) = 1/6.

Subtracting fractions with unlike denominators means the subtraction of fractions with the different denominator values. Go through the below steps to subtract the unlike fractions.

Step 1: Determine the LCM of the denominator values.

Step 2: Convert the denominator to the LCM value by multiplying the numerator and denominator using the same number.

Step 3: Subtract the numerators, once the fractions have the same denominator values.

Step 4: Simplify the fraction, if required.

Subtract 2/3 from 3/5.

Given: (3/5) – (2/3)

Find the LCM of 3 and 5. The LCM of 3 and 5 is 15.

To make the denominators equal, convert the denominators to the LCM value.

Thus, (3/5) – (2/3) = (9/15) – (10/15)

Now, the denominators are equal and we can subtract the numerator values.

(3/5) – (2/3) = (9/15) – (10/15)

= (9-10)/15

Therefore, (3/5) – (2/3) = -1/15.

While subtracting mixed fractions, go through the following steps:

Step 1: Convert mixed fractions into the improper fraction.

Step 2: Now, check the denominator values

• If the fractions are like fractions, follow the procedure of subtracting fractions with like denominators.
• If the fractions are unlike fractions, follow the procedure of subtracting fractions with unlike denominators.

Subtract 8 ⅚ from 15 ¾.

Given: (15 ¾) – (8 ⅚ )

Now, convert mixed fractions into improper fractions.

(15 ¾) – (8 ⅚ ) = (63/4)- (53/6)

Now, find the LCM of 4 and 6 and make the denominators equal.

Thus, LCM of 4 and 6 is 12

(63/4)- (53/6) = (189/12) – (106/12)

(63/4)- (53/6) = 83/12

Therefore, (15 ¾) – (8 ⅚ ) = 83/12

We can convert improper fractions into mixed fractions if required.

Follow the below steps while subtracting the fractions with whole numbers:

Step 1: Convert the whole number into the fractional form. For example, if 4 is a whole number, convert it into a fraction as 4/1

Step 2: Now, follow the procedure of subtracting fractions with unlike denominators.

Step 3: Simplify the fraction, if required.

Subtract: 2 – (½)

Given: 2- (½)

Convert the whole number “2” into the fractional form as “2/1”.

Therefore, 2 – (½) = (2/1)- (½)

Now, take the LCM of 1 and 2.

The LCM of 1 and 2 is 2.

(2/1) – (1/2) = (4/2) – (1/2)

(2/1) – (1/2) = (4-1)/2

(2/1) – (1/2) = 3/2

Therefore, 2 – (1/2) = 3/2.

Video Lesson on Fractions

Also, check : Subtracting Fractions Calculator .

Practice Problems on Subtracting Fractions

Solve the following problems related to subtracting fractions:

• Subtract ⅘ from 10/5.
• Subtract 9/2 from 11/3.
• Subtract: 5 – (4/3).
• Subtract: (6 ½) – (2 ⅘)
• Subtract: (2/7) – 4.

Frequently Asked Questions on Subtracting Fractions

What is meant by subtracting fractions.

In Mathematics, subtracting fractions means the process of subtraction of fractions with like or unlike denominators.

How do you subtract the fractions with the same denominators?

If the denominators of the fractions are the same, then subtract the numerators directly, and simplify the resulting fraction, if required.

How do you subtract the fractions with different denominators?

While subtracting fractions with different denominator, follow the steps given below: Step 1: Take the LCM of the denominator. Step 2: Convert the denominators to the LCM Value by multiplying the numerator and the denominator by the same number. Step 3: Once the denominator becomes equal, subtract the numerator. Step 4: Simplify the resulting fraction, if necessary.

Can we subtract fractions with whole numbers?

Yes, we can subtract fractions with whole numbers. Convert the whole number into fractions by dividing the whole number by 1 and perform the subtraction operation.

How do you subtract the mixed fractions?

While subtracting mixed fractions, convert the mixed fractions into the improper fraction and perform the subtraction operation. Finally, convert the result into a mixed fraction, if required.

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Subtracting Fractions Word Problems Worksheets

Thumb through our printable subtracting fractions word problems worksheets and discover a treasure of fun, realistic scenarios. Our pdf prepping tools, with included answer keys, are well-chosen for grade 3 through grade 6 students. Task young learners with reading, understanding, and solving an array of fraction subtraction word problems. Deftly find the difference between two like fractions, two unlike fractions, mixed numbers, and more. Go after our free subtracting fractions word problems worksheet!

Subtracting Fractions with Whole Numbers Word Problems

Perk up your math sessions with this section of our subtracting fractions word problems worksheets. Have kids determining the difference between fractions and whole numbers in real life with nuance and flair!

• Download the set

Subtracting Like Fractions Word Problems

Pride yourself on your subtraction on fractions skills! The interactive scenarios in our pdf word problems worksheets for 3rd grade, 4th grade, and 5th grade make subtracting like fractions a rewarding affair!

Subtracting Unlike Fractions Word Problems

Enhance problem-solving skills with these fraction subtraction word problems worksheets. Young learners read and solve word problems involving fractions with different denominators with accuracy.

Subtracting Mixed Numbers Word Problems | Like Denominators

Do nothing by halves! Get kids going all out to subtract mixed numbers having same denominators! The key to solving these problems is finding the difference between the whole number parts and fractions separately.

Subtracting Mixed Numbers Word Problems | Unlike Denominators

Add bells and whistles to your fraction subtraction practice with our printables. Grade 4, grade 5, and grade 6 kids find the LCM of different denominators and obtain the difference between two mixed numbers.

Related Worksheets

» Subtracting Like Fractions

» Subtracting Unlike Fractions

» Subtracting Mixed Numbers

» Subtracting Fractions from Whole Numbers

» Fraction Word Problems

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Add and Subtract Fractions Online practice for grades 3-7

On this page, you can practice addition and subtraction of fractions. Each practice set will automatically include both addition and subtraction problems.

The options are:

• You can limit the fractions in the problems to like fractions (fractions with the same denominator), for example: 1/6 + 4/6.
• You can limit the script to use only proper fractions—fractions that are less than 1. With this option, the script will make problems such as 1/4 + 2/5, but will not make problems such as 8/5 − 4/5.
• When you choose problems that use simplified fractions, the script will only include fractions in the problems that are in lowest terms. For example, you could get a problem such as 5/6 + 3/5, but you would not see 2/4 + 6/8.
• The last option, when chose, allows or accepts answers to not be in lowest terms. In other words, the script will accept an answer such as 8/10.

Note: ALL answers have to be given as mixed numbers, when possible. In other words, your answer cannot be left as an improper fraction.

You may use the space below to write the intermediate step. Your work in this area will not be checked.

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Addition and Subtraction of Fraction: Methods, Examples, Facts, FAQs

What is addition and subtraction of fractions, methods of addition and subtraction of fractions, addition and subtraction of mixed numbers, solved examples on addition and subtraction of fractions, practice problems on addition and subtraction of fractions, frequently asked questions on addition and subtraction of fractions.

Addition and subtraction of fractions are the fundamental operations on fractions that can be studied easily using two cases:

• Addition and subtraction of like fractions (fractions with same denominators)
• Addition and subtraction of unlike fractions (fractions with different denominators)

A fraction represents parts of a whole. For example, the fraction 37 represents 3 parts out of 7 equal parts of a whole. Here, 3 is the numerator and it represents the number of parts taken. 7 is the denominator and it represents the total number of parts of the whole.

Adding and subtracting fractions is simple and straightforward when it comes to like fractions. In the case of unlike fractions, we first need to make the denominators the same. Let’s take a closer look at both these cases.

Before adding and subtracting fractions, we first need to make sure that the fractions have the same denominators. 

When the denominators are the same, we simply add the numerators and keep the denominator as it is. To add or subtract unlike fractions, we first need to learn how to make the denominators alike. Let’s learn how to add fractions and how to subtract fractions in both cases.

Related Worksheets

Addition and Subtraction of Like Fractions

The rules for adding fractions with the same denominator are really simple and straightforward. 

Let’s learn with the help of examples and visual bar models.

Addition of Like Fractions

Here are the steps to add fractions with the same denominator:

Step 1: Add the numerators of the given fractions. 

Step 2: Keep the denominator the same. 

Step 3: Simplify.          

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$  …$c \neq 0$

Example 1: Find $\frac{1}{4} + \frac{2}{4}$ .

$\frac{1}{4} + \frac{2}{4} = \frac{1 + 2}{4} = \frac{3}{4}$

We can visualize this addition using a bar model:

Example 2: $\frac{1}{8} + \frac{3}{8} = \frac{1 + 3}{8} = \frac{4}{8} = \frac{1}{2}$

Subtraction of Like Fractions

Here are the steps to subtract fractions with the same denominator:

Step 1: Subtract the numerators of the given fractions. 

Step 3: Simplify. 

$\frac{a}{c}\;-\;\frac{b}{c} = \frac{a \;-\; b}{c}$ …$c \neq 0$

Example 1: Find $\frac{4}{6} \;-\; \frac{1}{6}$.

$\frac{4}{6}\;-\;\frac{1}{6} = \frac{4-1}{6} = \frac{3}{6} = \frac{1}{2}$

Addition and Subtraction of Unlike Fractions

Addition and subtraction of fractions with unlike denominators can be a little bit tricky since the denominators are not the same. So, we need to first convert the unlike fractions into like fractions. Let’s look at a few ways to do this!

Addition of Unlike Fractions

We can make the denominators the same by finding the LCM of the two denominators. Once we calculate the LCM, we multiply both the numerator and the denominator with an appropriate number so that we get the LCM value in the denominator. 

Example: $\frac{3}{5} + \frac{3}{2}$

Step 1: Find the LCM (Least Common Multiple) of the two denominators.

The LCM of 5 and 2 is 10.

Step 2: Convert both the fractions into like fractions by making the denominators same.  

$\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$  

$\frac{3 \times 5}{2 \times 5} = \frac{15}{10}$

Step 3: Add the numerators. The denominator stays the same.

$\frac{6}{10} + \frac{15}{10} = \frac{21}{10}$

Step 4: Convert the resultant fraction to its simplest form if the GCF of the numerator and denominator is not 1. 

In this case, GCF (21,10) $= 1$

The fraction $\frac{21}{10}$ is already in its simplest form. 

Thus, $\frac{3}{5} + \frac{3}{2} = \frac{21}{10}$

Subtraction of Unlike Fractions

Let’s learn how to subtract fractions when denominators are not the same. To subtract unlike fractions, we use the LCM method. The process is similar to what we discussed in the previous example.

Example: $\frac{5}{6} \;-\; \frac{2}{9}$

Step 1: Find the LCM of the two denominators.

LCM of 6 and $9 = 18$

Step 2: Convert both the fractions into like fractions by making the denominators same.

$\frac{5 \times 3}{6 \times 3} = \frac{15}{18}$   

$\frac{2 \times 2}{9 \times 2} = \frac{4}{18}$

Step 3: Subtract the numerators. The denominator stays the same.

$\frac{15}{18} \;-\; \frac{4}{18} = \frac{11}{18}$

In this case, the GCF (11,18) $= 1$

So, it is already in its simplest form. 

Thus, $\frac{5}{6}\;-\; 29 = \frac{11}{18}$

A mixed number is a type of fraction that has two parts: a whole number and a proper fraction. It is also known as a mixed fraction. Any mixed number can be written in the form of an improper fraction and vice-versa. 

Adding and subtracting mixed fractions is done by converting mixed numbers into improper fractions .

Addition and Subtraction of Mixed Fractions with Same Denominators

The steps of adding and subtracting mixed numbers with the same denominators are the same. The only difference is the operation.

Step 1: Convert the given mixed fractions to improper fractions.

Step 2: Add/Subtract the like fractions obtained in step 1.

Step 3: Reduce the fraction to its simplest form.

Step 4: Convert the resulting fraction into a mixed number.

Example 1: $2\frac{1}{5} + 1\frac{3}{5}$

$2\frac{1}{5} = \frac{(5 \times 2) + 1}{5} = \frac{11}{5}$

$1\frac{3}{5} = \frac{(5 \times 1) + 3}{5} = \frac{8}{5}$

Thus, $2\frac{1}{5} + 1\frac{3}{5} = \frac{11}{5} + \frac{8}{5} = \frac{19}{5}$

Converting $\frac{19}{5}$ into a mixed number, we get

$\frac{19}{5} = 3\frac{4}{5}$

Example 2: $2\frac{1}{5} + 1\frac{3}{5} = \frac{11}{5} \;-\; \frac{8}{5} = \frac{3}{5}$

Addition and Subtraction of Mixed Fractions with Unlike Denominators

Step 2: Convert both the fractions into like fractions by finding the least common denominator.

Step 3: Add the fractions. (or subtract the fractions.)

Step 4: Reduce the fraction if possible or convert back to a mixed number 

Let us understand the addition of mixed numbers with unlike denominators with the help of an example.

Example 1: Find the value of $1\frac{3}{5} + 2\frac{1}{2}$.

Convert the given mixed fractions to improper fractions.

$1\frac{3}{5} = \frac{8}{5}$ and $2\frac{1}{2} = \frac{5}{2}$

Step 2: Convert both the fractions into like fractions by making the denominators the same.

Here, LCM of 5 and 2 is 10.

Thus, $\frac{8 \times 2}{5 \times 2} = \frac{16}{10}$ and $\frac{5\times 5}{2 \times 5} = \frac{25}{10}$

Step 3: Add the fractions by adding the numerators.

$\frac{16}{10} + \frac{25}{10} = \frac{41}{10}$

Step 4: Convert back into a mixed number. 

Thus, $\frac{41}{10}$ will become  $4\frac{1}{10}$

Therefore, $1\frac{3}{5} + 2\frac{1}{2} =  4\frac{1}{10}$

Here’s an example for subtraction. It follows the same steps.

Example 2 : $6\frac{1}{2} \;-\; 1\frac{3}{4}$

Step 1: Convert the mixed numbers into improper fractions.

     $6\frac{1}{2} \;-\; 1\frac{3}{4} = \frac{13}{2} \;-\; \frac{7}{4}$

Step 2: Make the denominators equal.

LCM of 2 and 4 is 4. 

   $\frac{13 \times 2}{2 \times 2} = \frac{26}{4}$ 

Step 3: Subtract the fractions.

        $\frac{26}{4} \;-\;  \frac{7}{4} = \frac{19}{4}$

Step 4: Convert the fraction as a mixed number.

            $\frac{19}{4}  = 4\frac{3}{4}$  

Thus, $6\frac{1}{2} \;-\; 1\frac{3}{4}  =   4\frac{3}{4}$  

Facts about Addition and Subtraction of Fractions

• We cannot add or subtract fractions without converting them into like fractions.
• Like fractions are fractions that have the same denominator, and unlike fractions are fractions that have different denominators.
• Equivalent fractions are two different fractions that represent the same value.
• The LCD (least common denominator) of two fractions is the LCM of the denominators.

In this article, we have learned about addition and subtraction of fractions (like fractions, unlike fractions, mixed fractions), methods of addition and subtraction of these fractions along with the steps. Let’s solve some examples on adding and subtracting fractions to understand the concept better.

• Solve: $\frac{2}{4} + \frac{1}{4}$ .

Solution: 

Here, the denominators are the same.

Thus, we add the numerators by keeping the denominators as it is.

$\frac{2}{4} + \frac{1}{4} = \frac{2 + 1}{4}$ 

$\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

2. Find the sum of the fractions $\frac{3}{5}$ and $\frac{5}{2}$ by using the LCM method.

$\frac{3}{5}$ and $\frac{5}{2}$ are unlike fractions.

The LCM of 2 and 5 is 10.

Thus, we can write

$\frac{3}{5} + \frac{5}{2} = \frac{3 \times 2}{5 \times 2} + \frac{5 \times 5}{2 \times 5}$

$= \frac{6}{10} + \frac{25}{10}$

            $= \frac{6}{10} + \frac{25}{10}$

            $= \frac{31}{10}$

Thus, $\frac{3}{5} + \frac{5}{2} =  \frac{31}{10}$

3. Find $\frac{4}{16} + \frac{5}{8}$.

Solution:  

To add two fractions with different denominators, we first need to find the LCM of the denominators.

The LCM of 16 and 8 is 16.

$\frac{4}{16} + \frac{5}{8} = \frac{4 \times 1}{16\times 1} + \frac{5 \times 2}{8 \times 2}$ 

            $= \frac{10}{16} + \frac{4}{16}$ 

            $= \frac{14}{16}$

$= \frac{7}{8}$

4. From a rope $12\frac{1}{2}$ ft. long, a $7 \frac{6}{8}\;-$ ft-long piece is cut off. Find the length of the remaining rope.

Total length of the rope $= 12\frac{1}{2}$ ft.

Length of the rope that was cut off $= 7 \frac{6}{8}$ ft. 

The length of the remaining rope $= 12\frac{1}{2} \;-\; 7 \frac{6}{8}$

$12\frac{1}{2} \;-\; 7 \frac{6}{8} = \frac{25}{2} \;-\; \frac{62}{8}$

         $= \frac{25 \times 4}{2 \times 4} \;-\; \frac{62 \times 1}{8\times 1}$

         $= \frac{100}{8} \;-\; \frac{62}{8}$

         $= \frac{38}{8}$

         $= \frac{19}{4}$

Converting it into a mixed fraction, $\frac{19}{4}$ becomes $4 \frac{3}{4}$.

Thus, the length of the remaining rope is $4\frac{3}{4}$ ft.

Attend this quiz & Test your knowledge.

Find $\frac{2}{4} + \frac{2}{4}$.

$\frac{7}{24} + \frac{5}{16} =$, what is the least common denominator of $\frac{1}{2}$ and $\frac{1}{3}$, $\frac{3}{6} \;-\; \frac{1}{6} =$, what equation does the following figure represent.

How do we add and subtract negative fractions?

Negative fractions are simply fractions with a negative sign. The steps to add and subtract the negative fractions remain the same. We need to follow the rules for addition/subtraction with negative signs.

How can we convert an improper fraction into a mixed number?

To convert an improper fraction into a mixed number, we divide the numerator by the denominator. The denominator stays the same. The quotient represents the whole number part. The remainder represents the numerator of the mixed number.

Example: $\frac{14}{3} = 4\; \text{R}\; 2$

Quotient $= 4$

Remainder $= 2$

$\frac{14}{3} = 4\frac{2}{3}$

How do we divide two fractions?

To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction.

$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$

For example, $\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{5}{6}$

What are the rules of adding and subtracting fractions?

• Before adding or subtracting, we check if the fractions have the same denominator.
• If the denominators are equal, then we add/subtract the numerators keeping the common denominator.
• If the denominators are different, then we make the denominators equal by using the LCM method. Once the fractions have the same denominator, we can add/subtract the numerators keeping the common denominator as it is.

How do we add and subtract fractions with whole numbers?

• Convert the whole number to a fraction. To do this, give the whole number a denominator of 1.
• Convert to fractions of like denominators. 
• Add/subtract the numerators. Now that the fractions have the same denominators, you can treat the numerators as a normal addition/subtraction problem.

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2.3.2: Subtracting Fractions and Mixed Numbers

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• The NROC Project

Learning Objectives

• Subtract fractions with like and unlike denominators.
• Subtract mixed numbers without regrouping.
• Subtract mixed numbers with regrouping.
• Solve application problems that require the subtraction of fractions or mixed numbers.

Introduction

Sometimes subtraction, rather than addition, is required to solve problems that involve fractions. Suppose you are making pancakes and need \(\ 4 \frac{1}{2}\) cups of flour but you only have \(\ 2 \frac{3}{4}\) cups. How many additional cups will you have to get to make the pancakes? You can solve this problem by subtracting the mixed numbers.

Subtracting Fractions

The most simple fraction subtraction problems are those that have two proper fractions with a common denominator . That is, each denominator is the same. The process is just as it is for addition of fractions with like denominators , except you subtract! You subtract the second numerator from the first and keep the denominator the same.

Imagine that you have a cake with equal-sized pieces. Some of the cake has already been eaten, so you have a fraction of the cake remaining. You could represent the cake pieces with the picture below.

The cake is cut into 12 equal pieces to start. Two are eaten, so the remaining cake can be represented with the fraction \(\ \frac{10}{12}\). If 3 more pieces of cake are eaten, what fraction of the cake is left? You can represent that problem with the expression \(\ \frac{10}{12}-\frac{3}{12}\).

If you subtract 3 pieces, you can see below that \(\ \frac{7}{12}\) of the cake remains.

You can solve this problem without the picture by subtracting the numerators and keeping the denominator the same:

\(\ \frac{10}{12}-\frac{3}{12}=\frac{7}{12}\)

Subtracting Fractions with Like Denominators

If the denominators (bottoms) of the fractions are the same, subtract the numerators (tops) and keep the denominator the same. Remember to simplify the resulting fraction, if possible.

\(\ \frac{6}{7}-\frac{1}{7}=\frac{5}{7}\)

\(\ \frac{5}{9}-\frac{2}{9}=\frac{1}{3}\)

If the denominators are not the same (they have unlike denominators ), you must first rewrite the fractions with a common denominator. The least common denominator , which is the least common multiple of the denominators, is the most efficient choice, but any common denominator will do. Be sure to check your answer to be sure that it is in simplest form. You can use prime factorization to find the least common multiple (LCM), which will be the least common denominator (LCD). See the example below.

\(\ \frac{1}{5}-\frac{1}{6}=\frac{1}{30}\)

The example below shows using multiples to find the least common multiple, which will be the least common denominator.

\(\ \frac{5}{6}-\frac{1}{4}=\frac{7}{12}\)

\(\ \frac{2}{3}-\frac{1}{6}\) Subtract and simplify the answer.

• \(\ \frac{1}{3}\)
• \(\ \frac{3}{6}\)
• \(\ \frac{5}{6}\)
• \(\ \frac{1}{2}\)
• Incorrect. Find a least common denominator and subtract; then simplify. The correct answer is \(\ \frac{1}{2}\).
• Incorrect. Simplify the fraction. The correct answer is \(\ \frac{1}{2}\).
• Incorrect. Subtract, don’t add, the fractions. The correct answer is \(\ \frac{1}{2}\)
• Correct. \(\ \frac{4}{6}-\frac{1}{6}=\frac{3}{6}=\frac{1}{2}\)

Subtracting Mixed Numbers

Subtracting mixed numbers works much the same way as adding mixed numbers. To subtract mixed numbers, subtract the whole number parts of the mixed numbers and then subtract the fraction parts in the mixed numbers. Finally, combine the whole number answer and the fraction answer to express the answer as a mixed number.

\(\ 6 \frac{4}{5}-3 \frac{1}{5}=3 \frac{3}{5}\)

Sometimes it might be easier to express the mixed number as an improper fraction first and then solve. Consider the example below.

\(\ 8 \frac{1}{3}-4 \frac{2}{3}=3 \frac{2}{3}\)

Since addition is the inverse operation of subtraction, you can check your answer to a subtraction problem with addition. In the example above, if you add \(\ 4 \frac{2}{3}\) to your answer of \(\ 3 \frac{2}{3}\), you should get \(\ 8 \frac{1}{3}\).

\(\ \begin{array}{r} 4 \frac{2}{3}+3 \frac{2}{3} \\ 4+3+\frac{2}{3}+\frac{2}{3} \\ 7+\frac{4}{3} \\ 7+1 \frac{1}{3} \\ 8 \frac{1}{3} \end{array}\)

Sometimes you have to find a common denominator in order to solve a mixed number subtraction problem.

\(\ 7 \frac{1}{2}-2 \frac{1}{3}=5 \frac{1}{6}\)

\(\ 9 \frac{4}{5}-4 \frac{2}{3}\)

Subtract. Simplify the answer and write it as a mixed number.

• \(\ \frac{2}{15}\)
• \(\ 5 \frac{2}{15}\)
• \(\ 4 \frac{7}{15}\)
• Incorrect. Subtract the whole numbers, too. The correct answer is \(\ 5 \frac{2}{15}\).
• Correct. \(\ 9-4=5\); \(\ \frac{4}{5}-\frac{2}{3}=\frac{12}{15}-\frac{10}{15}=\frac{2}{15}\). Combining them gives \(\ 5 \frac{2}{15}\).
• Incorrect. Subtract, don’t add, the fractions. The correct answer is \(\ 5 \frac{2}{15}\).
• Incorrect. Subtract the fractions as well as the whole numbers. The correct answer is \(\ 5 \frac{2}{15}\).

Subtracting Mixed Numbers with Regrouping

Sometimes when subtracting mixed numbers, the fraction part of the second mixed number is larger than the fraction part of the first number. Consider the problem: \(\ 7 \frac{1}{6}-3 \frac{5}{6}\). The standard procedure would be to subtract the fractions, but \(\ \frac{1}{6}-\frac{5}{6}\) would result in a negative number. You don’t want that! You can regroup one of the whole numbers from the first number, writing the first mixed number in a different way:

\(\ \begin{array}{l} 7 \frac{1}{6}=7+\frac{1}{6}=6+1+\frac{1}{6} \\ 6+\frac{6}{6}+\frac{1}{6}=6+\frac{7}{6}=6 \frac{7}{6} \end{array}\)

Now, you can write an equivalent problem to the original:

\(\ 6 \frac{7}{6}-3 \frac{5}{6}\)

Then, you just subtract like you normally subtract mixed numbers:

\(\ 6-3=3\)

\(\ \frac{7}{6}-\frac{5}{6}=\frac{2}{6}=\frac{1}{3}\)

So, the answer is \(\ 3 \frac{1}{3}\).

As with many fraction problems, you may need to find a common denominator. Remember that a key part of adding and subtracting fractions and mixed numbers is making sure to have a common denominator as a first step! In the example below, the original fractions do not have a like denominator. You need to find one before proceeding with the next steps.

\(\ 7 \frac{1}{5}-3 \frac{1}{4}=3 \frac{19}{20}\)

Sometimes a mixed number is subtracted from a whole number. In this case, you can also rewrite the whole number as a mixed number in order to perform the subtraction. You use an equivalent mixed number that has the same denominator as the fraction in the other mixed number.

\(\ 8-4 \frac{2}{5}=3 \frac{3}{5}\)

If the fractional part of the mixed number being subtracted is larger than the fractional part of the mixed number from which it is being subtracted, or if a mixed number is being subtracted from a whole number, follow these steps:

• Subtract 1 from the whole number part of the mixed number being subtracted.
• Add that 1 to the fraction part to make an improper fraction. For example: \(\ 7 \frac{2}{3}=6+\frac{3}{3}+\frac{2}{3}=6 \frac{5}{3}\)
• Then, subtract as with any other mixed numbers.

Alternatively, you can change both numbers to improper fractions and then subtract.

\(\ 15-13 \frac{1}{4}\) Subtract. Simplify the answer and write as a mixed number.

• \(\ 2 \frac{1}{4}\)
• \(\ 28 \frac{1}{4}\)
• \(\ 1 \frac{3}{4}\)
• \(\ 2 \frac{3}{4}\)
• Incorrect. This is the answer to \(\ 15 \frac{1}{4}-13\). The fraction has to be subtracted from the 15. The correct answer is \(\ 1 \frac{3}{4}\).
• Incorrect. Subtract, don’t add, the quantities. The correct answer is \(\ 1 \frac{3}{4}\).
• Correct. \(\ 14 \frac{4}{4}-13 \frac{1}{4}=1 \frac{3}{4}\)
• Incorrect. Subtract 1 from the whole number when rewriting it as a mixed number. The correct answer is \(\ 1 \frac{3}{4}\).

Subtracting Fractions and Mixed Numbers to Solve Problems

Knowing how to subtract fractions and mixed numbers is useful in a variety of situations. When reading problems, look for key words that indicate that the problem can be solved using subtraction.

Sherry loves to quilt, and she frequently buys fabric she likes when she sees it. She had purchased 5 yards of blue print fabric and decided to use \(\ 2 \frac{3}{8}\) yards of it in a quilt. How much of the blue print fabric will she have left over after making the quilt?

Sherry has \(\ 2 \frac{5}{8}\) yards of blue print fabric left over.

Pilar and Farouk are training for a marathon. On a recent Sunday, they both completed a run. Farouk ran \(\ 12 \frac{7}{8}\) miles and Pilar ran \(\ 14 \frac{3}{4}\) miles. How many more miles did Pilar run than Farouk?

Pilar ran \(\ 1 \frac{7}{8}\) miles more than Farouk.

Mike and Jose are painting a room. Jose used \(\ \frac{2}{3}\) of a can of paint and Mike used \(\ \frac{1}{2}\) of a can of paint. How much more paint did Jose use? Write the answer as a fraction of a can.

Jose used \(\ \frac{1}{6}\) of a can more paint than Mike.

Mariah’s sunflower plant grew \(\ 18 \frac{2}{3}\) inches in one week. Her tulip plant grew \(\ 3 \frac{3}{4}\) inches in one week. How many more inches did the sunflower grow in a week than the tulip?

• \(\ 22 \frac{5}{12}\) inches
• \(\ 15 \frac{1}{12}\) inches
• \(\ 15\) inches
• \(\ 14 \frac{11}{12}\) inches
• Incorrect. Subtract, don’t add, the fractions. The correct answer is \(\ 14 \frac{11}{12}\) inches.
• Incorrect. Subtract \(\ \frac{2}{3}-\frac{3}{4}\), not \(\ \frac{3}{4}-\frac{2}{3}\). The correct answer is \(\ 14 \frac{11}{12}\) inches.
• Incorrect. Subtract the fractions as well as the whole numbers in the mixed numbers. The correct answer is \(\ 14 \frac{11}{12}\) inches.
• Correct. \(\ 17 \frac{20}{12}-3 \frac{9}{12}=14 \frac{11}{12}\)

Subtracting fractions and mixed numbers combines some of the same skills as adding whole numbers and adding fractions and mixed numbers. When subtracting fractions and mixed numbers, first find a common denominator if the denominators are not alike, rewrite each fraction using the common denominator, and then subtract the numerators. When subtracting mixed numbers, if the fraction in the second mixed number is larger than the fraction in the first mixed number, rewrite the first mixed number by regrouping one whole as a fraction. Alternatively, rewrite all fractions as improper fractions and then subtract. This process is also used when subtracting a mixed number from a whole number.

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Algebraic Fractions

You have encountered fractions many times since early in the study of mathematics. They occur in formulas and in many day-to-day practical problems. However, the fractions of arithmetic are made up strictly of numbers. We will now study operations on fractions whose components are algebraic expressions.

SOLVING EQUATIONS INVOLVING SIGNED NUMBERS

• Factor the numerator and denominator of a fraction.
• Simplify algebraic fractions.
• An algebraic fraction is the indicated ratio of two algebraic expressions.

A fraction is in simplified (or reduced) form if the numerator and denominator contain no common factor (other than 1).

To obtain the simplified form of a fraction apply the following rule.

To simplify a fraction factor the numerator and denominator completely and then divide both numerator and denominator by all common factors.

Next divide by the common factors, giving

Now divide by the common factor (x + 2) in both numerator and denominator to get

Notice that even though we were able to factor the numerator and denominator, we still cannot divide since no factors are common to both. The given fraction is already in simplified form.

The fact that a given fraction might require any of the methods of factoring you have studied emphasizes again the importance of being proficient in factoring.

Solution Here you may use "trial and error" for the numerator and "grouping" for the denominator.

Solution This type of problem requires special attention because it is a common cause for error. At first glance the factors might be mistakenly considered as common, or the fraction might be mistakenly considered as already simplified. Note that the factors cannot be divided since the signs keep them from being identical. If, however, negative 1 is factored from one of the factors, then there are like factors and division can be accomplished.

MULTIPLICATION OF ALGEBRAIC FRACTIONS

• Factor numerators and denominators of all fractions being multiplied.
• Identify and divide by all common factors.
• Write the product in simplest form.

However, remember that all fractional answers must be in simplified form. We could follow the above definition and then simplify the answer as in the previous section. But with algebraic fractions this can lead to very difficult expressions. The following rule allows us to simplify as we multiply, so the answer will then be in simplified form.

When multiplying algebraic fractions , factor all numerators and denominators completely, then divide by all factors common to a numerator and denominator before multiplying.

The product of the remaining factors of the numerator will be the numerator of the answer and the product of the remaining factors of the denominator will be the denominator of the answer.

DIVISION OF ALGEBRAIC FRACTIONS

• Change a division problem to a related multiplication problem.
• Divide algebraic fractions.

Division of fractions is defined in terms of multiplication.

To divide multiply by the inverse of the divisor.

To divide one algebraic expression by another invert the divisor and change the operation to multiplication.

After the problem is changed from a division problem to a multiplication problem, it is completed as in the previous section.

FINDING THE LEAST COMMON DENOMINATOR

• Factor a denominator of a fraction completely.
• Find the least common denominator of two or more fractions.

The rule for addition and subtraction of fractions requires that the fractions to be combined must have the same denominator. As preparation for performing these operations we will now investigate the method of finding the least common denominator for any group of fractions.

A common denominator lot two or more fractions is an expression that contains all factors of the denominator of each fraction. A least common denominator contains the minimum number of factors to be a common denominator.

Solution This problem would require a considerable amount of guesswork, or testing possibilities, if we had no general method.

Let's consider the definition. From it we know that a common denominator for these numbers must contain all factors of each. In other words, we are looking for the smallest number divisible by 12, 14, 15, and 18. First completely factor each number.

The number we are looking for must contain (2)(2)(3) in order to be divisible by 12. It must contain (2)(7) in order to be divisible by 14, and so on. Proceed as follows: Write the factors of the first number, 12. (2)(2)(3) Now look at the factors of the next number, 14, and see that we need (2)(7). But since we already have a 2, we only need the factor (7). This gives (2)(2)(3)(7). This number is now divisible by 12 and by 14. The factors of the next number, 15, are (3) and (5). Since we already have a 3, we need only the factor 5, giving (2)(2)(3)(7)(5). This number is now divisible by 12, 14, and 15. The factors of the next number, 18, are (2)(3)(3). We already have 2 and one 3. Hence, we need another 3. (2)(2)(3)(7)(5)(3) = 1,260 This number, 1,260, is a common denominator of 12, 14, 15, and 18 because it contains all factors of each and is therefore divisible by each. It is the least common denominator because it contains only those factors necessary to make it divisible by 12, 14, 15, and 18.

The preceding discussion gives rise to a rule for obtaining a least common denominator for any number of fractions, whether they be numbers or algebraic expressions.

To find the least common denominator for two or more fractions: 1. Factor each denominator completely. 2. Write the denominator of the first fraction in factored form as the proposed common denominator. 3. By inspection determine which factors of the second denominator are not already in the proposed common denominator and include them. 4. Repeat step three for each fraction.

By inspection of the second denominator we need the additional factor (x - 2). The least common denominator is (3x - 4)(2x + l)(x - 2).

Solution First denominator: 3(x + 2) Second denominator: 2(2)(3) Third denominator: 2(x + 3)(x + 2) Proposed common denominator: 3(x + 2) By inspection of the second denominator we see we need to include the factors (2) and (2). We now have 2(2)(3)(x + 2). By inspection of the third denominator we see we need the factor (x + 3). The least common denominator is 2(2)(3)(x + 2)(x + 3) or 12(x + 2)(x + 3).

EQUIVALENT FRACTIONS

• Understand the fundamental principle of fractions.
• Change a fraction to an equivalent fraction.

In further preparation for adding and subtracting fractions, we must be able to change a given fraction to one with a new denominator without changing the value of the original fraction.

• To change a fraction to an equivalent fraction multiply numerator and denominator by the same nonzero expression.

Solution Since the new denominator is in factored form, by inspection we see that the original denominator (2x + 3) has been multiplied by the factor (x - 4). Therefore, the original numerator (x + 1) must also be multiplied by the factor (x - 4), giving

Solution Since the original denominator (x - 3) has been multiplied by (2) and (x + 1), the original numerator (2x + 1) must also be multiplied by (2) and (x + 1).

ADDITION OF ALGEBRAIC FRACTIONS

• Add fractions having the same denominator.
• Apply the rule for adding fractions.

We are now prepared to add algebraic fractions by using the techniques discussed in the preceding two sections. You should recall the following rule from arithmetic.

The sum of two or more fractions which have the same denominator is the sum of the numerators over their common denominator.

Take note that this rule only allows the sum of fractions that have the same denominator. In other words, two or more fractions can only be added if they have a common denominator. The rule for adding any two or more fractions will require the skills developed in the last two sections in addition to knowledge of combining like terms.

To add two or more fractions follow these steps: Step 1 Find the least common denominator (LCD) for all fractions involved using the method developed in section 9-4. Step 2 Change each fraction to an equivalent fraction having the least common denominator (section 9-5). Step 3 Find the sum of the numerators and place this sum over the least common denominator. Step 4 Simplify (or reduce) the fraction obtained in step 3.

This answer is in reduced form.

We can use fewer written steps if we note that "common denominator" means all the fractions have the same denominator, and if all have the same denominator, then it is necessary to write the denominator only once. To illustrate this we will re-work the preceding example.

SUBTRACTION OF ALGEBRAIC FRACTIONS

• Subtract fractions having the same denominator.
• Apply the rule for subtracting fractions with unlike denominators.

Subtraction is defined in terms of addition, so the method of subtracting algebraic fractions will be the same as adding algebraic fractions discussed in the preceding section. You will soon see why we have presented them separately.

The difference of any two fractions having the same denominator is the difference of their numerators over their common denominator.

Notice that this rule is the same as the rule for adding two fractions with the same denominator.

The steps for subtracting fractions are, therefore, the same as for adding fractions.

To subtract fractions: Step 1 Find the least common denominator of the two fractions. Step 2 Change each fraction to an equivalent fraction having the least common denominator. Step 3 Find the difference of the numerators and place this result over the least common denominator. Step 4 Simplify (or reduce) the fraction obtained in step 3.

The obvious question is, "If these two operations are the same, why study them separately?" The answer is that the subtraction gives rise to a very common error that the student must be prepared to avoid.

The error referred to is often made by not recognizing that the minus sign affects the entire numerator of the second fraction and NOT just the first term.

The arrow points out the error most commonly made in subtraction of fractions. The best way to avoid this is to always use parentheses

and you are not so likely to fail to change the sign properly.

COMPLEX FRACTIONS

• Recognize a complex fraction.
• Simplify a complex fraction.

Fractions are defined as the indicated quotient of two expressions. In this section we will present a method for simplifying fractions in which the numerator or denominator or both are themselves composed of fractions. Such fractions are called complex fractions .

Thus if the numerator and denominator of a complex fraction are composed of single fractions, it can be simplified by dividing the numerator by the denominator.

A generally more efficient method of simplifying a complex fraction involves using the fundamental principle of fractions. We multiply both numerator and denominator by the common denominator of all individual fractions in the complex fraction.

EQUATIONS HAVING ALGEBRAIC FRACTIONS

• Apply the method of solving fractional equations.
• Determine when a fractional equation does not have a solution.

In chapter 2 we encountered equations that have fractions. However, those fractions all had numerical denominators. Now we will discuss equations that have fractions involving variables in the denominators.

The method of solving these equations will follow the same pattern as in chapter 2, but there are some additional cautions that you must be prepared to take.

To refresh your memory the steps for solving such equations are repeated here. First: Eliminate fractions by multiplying each term of the equation by the least common denominator of all fractions in the equation. Second: Simplify by combining like terms on each side of the equation. Third: Add or subtract the necessary quantities to get the unknown quantity on one side and the numbers of arithmetic on the other. Fourth: Divide by the coefficient of the unknown quantity. Fifth: Check your answer.

The main difference in solving equations with arithmetic fractions and those with algebraic fractions comes in checking. The checking process will not just be to find a possible error, but will also be to determine if the equation has an answer.

This last possibility arises because with algebraic fractions we multiply by an unknown quantity. This unknown quantity could actually be zero, which would make all the work invalid.

Since division by zero is not possible, we must conclude that x = 1 is not a solution. And since we made no error in the computations we must conclude that this equation has no solution. The correct answer is "no solution."

Thus, x = -5 is a solution.

Therefore, 11 is the amount by which the numerator was increased.

• A fraction is in simplified form if the numerator and denominator have no common factor other than 1.
• A common denominator for two or more fractions is an expression that contains all factors of the denominators of each fraction.
• A least common denominator contains the minimum number of factors to be a common denominator.

• Complex fractions are those in which the numerator or denominator (or both) contains a fraction.
• To simplify or reduce fractions to lowest terms factor the numerator and denominator and divide by all like factors.
• To multiply fractions factor all numerators and denominators and divide by all like factors before multiplying.
• To divide by a fraction invert the divisor and then multiply.
• To find a least common denominator (LCD) first factor all denominators, then find a denominator that contains all factors of each denominator but does not contain any unnecessary factors.
• Find the least common denominator.
• Change each fraction to an equivalent fraction having the LCD as its denominator.
• Add the numerators and place over the LCD.
• Simplify or reduce the answer.
• To subtract fractions proceed as in addition but combine the numerators by subtracting.
• Complex fractions may be simplified by multiplying numerator and denominator of the complex fraction by the LCD of all fractions in the expression.
• To solve equations having fractions first eliminate all fractions by multiplying the entire equation by the LCD of the fractions involved. The resulting equation is then solved, and the solution must be checked in the original equation.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

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Unit 4: Understand fractions

About this unit.

From cooking measurements to geometry, fractions are all around us. By understanding how the numerator and denominator work together, you'll be able to break down numbers into smaller parts, compare different fractions, and get a grasp on concepts like equivalent fractions.

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Equivalent fractions

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Adding and Subtracting Fractions

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Adding and Subtracting Fractions with Like Denominators Worksheets

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• Adding Fractions with Like Denominators
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