problem solving involving subtraction of decimals

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Subtracting Decimals Word Problems Worksheets

• Pre-Algebra >
• Decimals >
• Subtraction >

Be streets ahead of your peers with our free worksheets on subtracting decimals word problems, replete with realistic scenarios where decimal subtraction is center stage! Children are expected to read each problem and find the difference between the decimals. Chances are the budding mathematicians have already perfected the skill of subtracting decimals, so invite them to demonstrate their problem-solving skills in these printable tools and to impress by finding quick solutions to situations featuring weight, height, money, distance, and more. Our word problems on decimal subtraction are available in both customary and metric units.

We suggest these pdf worksheets for the 4th grade, 5th grade, and 6th grade students.

Related Printable Worksheets

▶ Subtracting Decimals with Tenths

▶ Subtracting Decimals with Hundredths

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6.4: Addition and Subtraction of Decimals

• Last updated
• Save as PDF
• Page ID 48867

• Denny Burzynski & Wade Ellis, Jr.
• College of Southern Nevada via OpenStax CNX

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Learning Objectives

• understand the method used for adding and subtracting decimals
• be able to add and subtract decimals
• be able to use the calculator to add and subtract decimals

The Logic Behind the Method

Consider the sum of 4.37 and 3.22. Changing each decimal to a fraction, we have

\(4 \dfrac{37}{100} + 3 \dfrac{22}{100}\) Performing the addition, we get

\(\begin{array} {rcl} {4.37 + 3.22 = 4 \dfrac{37}{100} + 3 \dfrac{22}{100}} & = & {\dfrac{4 \cdot 100 + 37}{100} + \dfrac{3 \cdot 100 + 22}{100}} \\ {} & = & {\dfrac{437}{100} + \dfrac{322}{100}} \\ {} & = & {\dfrac{437 + 322}{100}} \\ {} & = & {\dfrac{759}{100}} \\ {} & = & {7 \dfrac{59}{100}} \\ {} & = & {\text{seven and fifty-nine hundredths}} \\ {} & = & {7.59} \end{array}\)

Thus, \(4.37 + 3.22 = 7.59\).

The Method of Adding and Subtracting Decimals

When writing the previous addition, we could have written the numbers in col­umns.

\(\begin{array} {r} {4.37} \\ {\underline{+3.22}} \\ {7.59} \end{array}\)

This agrees with our previous result. From this observation, we can suggest a method for adding and subtracting decimal numbers.

Method of Adding and Subtracting Decimals To add or subtract decimals:

Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column. Add or subtract the numbers as if they were whole numbers. Place a decimal point in the resulting sum or difference directly under the other decimal points.

Sample Set A

Find the following sums and differences.

\(9.813 + 2.140\)

\(\begin{array} {r} {9.813} \\ {\underline{+2.140}} \\ {11.953} \end{array}\) The decimal points are aligned in the same column.

\(841.0056 + 47.016 + 19.058\)

\(\begin{array} {r} {841.0056} \\ {47.016\ \ } \\ {\underline{+19.058\ \ }} \end{array}\)

To insure that the columns align properly, we can write a 0 in the position at the end of the numbers 47.016 and 19.058 without changing their values.

\(1.314 - 0.58\)

\(\begin{array} {r} {1.314} \\ {\underline{-0.58\ \ }} \end{array}\) Write a 0 in the thousandths position.

\(16.01 - 7.053\)

\(\begin{array} {r} {16.01\ \ } \\ {\underline{-7.053}} \end{array}\) Write a 0 in the thousandths position.

Find the sum of 6.88106 and 3.5219 and round it to three decimal places.

\(\begin{array} {r} {6.88106} \\ {\underline{+3.5219\ \ }} \end{array}\) Write a 0 in the ten thousandths position.

We need to round the sum to the thousandths position. Since the digit in the position immediately to the right is 9, and 9>5, we get

Wendy has $643.12 in her checking account. She writes a check for $16.92. How much is her new account balance?

To find the new account balance, we need to find the difference between 643.12 and 16.92. We will subtract 16.92 from 643.12.

After writing a check for $16.92, Wendy now has a balance of $626.20 in her checking account.

Pracitce Set A

\(3.187 + 2.992\)

\(14.987 - 5.341\)

\(0.5261 + 1.0783\)

\(1.06 - 1.0535\)

\(16,521.07 + 9,256.15\)

Find the sum of 11.6128 and 14.07353, and round it to two decimal places.

Calculators

The calculator can be useful for finding sums and differences of decimal numbers. However, calculators with an eight-digit display cannot be used when working with decimal numbers that contain more than eight digits, or when the sum results in more than eight digits. In practice, an eight-place decimal will seldom be encoun­tered. There are some inexpensive calculators that can handle 13 decimal places.

Sample Set B

Use a calculator to find each sum or difference.

42.0638 + 126.551

The sum is 168.6148.

Find the difference between 305.0627 and 14.29667.

The difference is 290.76603

51.07 + 3,891.001786

Since 3,891.001786 contains more than eight digits, we will be unable to use an eight-digit display calculator to perform this addition. We can, however, find the sum by hand.

\(\begin{array} {r} {51.070000} \\ {\underline{3891.001786}} \\ {3942.071786} \end{array}\)

The sum is 3,942.071786.

Practice Set B

Use a calculator to perform each operation.

\(4.286 + 8.97\)

\(452.0092 - 392.558\)

Find the sum of 0.095 and 0.001862

Find the difference between 0.5 and 0.025

Find the sum of 2,776.00019 and 2,009.00012.

Since each number contains more than eight digits, using some calculators may not be helpful. Adding these by “hand technology,” we get 4,785.00031

For the following 15 problems, perform each addition or subtraction. Use a calculator to check each result.

Exercise \(\PageIndex{1}\)

\(1.84 + 7.11\)

Exercise \(\PageIndex{2}\)

\(15.015 - 6.527\)

Exercise \(\PageIndex{3}\)

\(11.842 + 28.004\)

Exercise \(\PageIndex{4}\)

\(3.16 - 2.52\)

Exercise \(\PageIndex{5}\)

\(3.55267 + 8.19664\)

Exercise \(\PageIndex{6}\)

\(0.9162 - 0.0872\)

Exercise \(\PageIndex{7}\)

\(65.512 - 8.3005\)

Exercise \(\PageIndex{8}\)

\(761.0808 - 53.198\)

Exercise \(\PageIndex{9}\)

\(4.305 + 2.119 - 3.817\)

Exercise \(\PageIndex{10}\)

\(19.1161 + 27.8014 + 39.3161\)

Exercise \(\PageIndex{11}\)

\(0.41276 - 0.0018 - 0.00011\)

Exercise \(\PageIndex{12}\)

\(2.181 + 6.05 + 1.167 + 8.101\)

Exercise \(\PageIndex{13}\)

\(1.0031 + 6.013106 + 0.00018 + 0.0092 + 2.11\)

Exercise \(\PageIndex{14}\)

\(27 + 42 + 9.16 - 0.1761 + 81.6\)

Exercise \(\PageIndex{15}\)

\(10.28 + 11.111 + 0.86 + 5.1\)

For the following 10 problems, solve as directed. A calculator may be useful.

Exercise \(\PageIndex{16}\)

Add 6.1121 and 4.916 and round to 2 decimal places.

Exercise \(\PageIndex{17}\)

Add 21.66418 and 18.00184 and round to 4 decimal places.

Exercise \(\PageIndex{18}\)

Subtract 5.2121 from 9.6341 and round to 1 decimal place.

Exercise \(\PageIndex{19}\)

Subtract 0.918 from 12.006 and round to 2 decimal places.

Exercise \(\PageIndex{20}\)

Subtract 7.01884 from the sum of 13.11848 and 2.108 and round to 4 decimal places.

Exercise \(\PageIndex{21}\)

A checking account has a balance of $42.51. A check is written for $19.28. What is the new balance?

Exercise \(\PageIndex{22}\)

A checking account has a balance of $82.97. One check is written for $6.49 and another for $39.95. What is the new balance?

Exercise \(\PageIndex{23}\)

A person buys $4.29 worth of hamburger and pays for it with a $10 bill. How much change does this person get?

Exercise \(\PageIndex{24}\)

A man buys $6.43 worth of stationary and pays for it with a $20 bill. After receiving his change, he realizes he forgot to buy a pen. If the total price of the pen is $2.12, and he buys it, how much of the $20 bill is left?

Exercise \(\PageIndex{25}\)

A woman starts recording a movie on her video cassette recorder with the tape counter set at 21.93. The movie runs 847.44 tape counter units. What is the final tape counter reading?

Exercises for Review

Exercise \(\PageIndex{26}\)

Find the difference between 11,206 and 10,884.

Exercise \(\PageIndex{27}\)

Find the product, \(820 \cdot 10,000\).

Exercise \(\PageIndex{28}\)

Find the value of \(\sqrt{121} - \sqrt{25} + 8^2 + 16 \div 2^2\).

Exercise \(\PageIndex{29}\)

Find the value of \(8 \dfrac{1}{3} \cdot \dfrac{36}{75} \div 2 \dfrac{2}{5}\).

\(\dfrac{20}{9} = \dfrac{5}{3}\) or \(2 \dfrac{2}{9}\)

Exercise \(\PageIndex{30}\)

Round 1.08196 to the nearest hundredth.

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How to Perform Operations of Decimals: Word Problems

Greetings, budding mathematicians! Today, we're going to jump into an adventure with decimals. Working with decimals is a fundamental math skill that you'll use in many aspects of life, from managing money to measuring distances.

Introduction to Decimal Operations

First things first, let’s talk about decimals. A decimal is a way of representing a number that’s less than one, or a number that’s a whole and a fraction together. When you work with decimals, the same rules apply as when you’re working with whole numbers, but the decimal point plays a crucial role.

There are four major operations that you can perform on decimals:

• Subtraction
• Multiplication

Now, let’s tackle some word problems involving decimal operations.

Step-By-Step Guide to Solving Word Problems with Decimal Operations

Step 1: understand the problem.

First and foremost, carefully read through the problem. Identify what you know and what you need to find out.

Step 2: Plan the Solution

Next, decide which operation (addition, subtraction, multiplication, or division) will help you solve the problem. This will depend on what the problem is asking.

Step 3: Carry Out the Operation

Perform the operation on the decimals. Remember to align the decimal points when you’re adding or subtracting, and apply the rules for multiplication and division of decimals correctly.

Step 4: Check the Answer

Finally, check if your answer makes sense in the context of the problem.

Consider this problem: Laura bought \(2.3\) kilograms of apples and \(1.5\) kilograms of grapes. How many kilograms of fruit did she buy in total?

Step 1: Understand that you know the weight of apples and grapes separately, and you need to find the total weight.

Step 2: Realize that to find the total, you need to add the weights together.

Step 3: Add the weights: \(2.3\ kg + 1.5\ kg = 3.8\ kg\).

Step 4: Check that the answer makes sense. Laura bought \(2.3\ kg\) of apples and \(1.5\ kg\) of grapes, so it makes sense that she bought \(3.8\ kg\) in total.

Keep practicing with different word problems, and you’ll get the hang of decimal operations in no time. Remember, the more you practice, the better you’ll get. Happy calculating!

by: Effortless Math Team about 10 months ago (category: Articles )

Effortless Math Team

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Mastering Grade 6 Math Word Problems The Ultimate Guide to Tackling 6th Grade Math Word Problems

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Decimal Word Problems (1-Step Word Problem)

In these lessons, we will learn to use block models (or tape diagrams, bar models) to visualize and solve decimal work problems.

Related Pages More Word Problems Decimal Worksheets More Singapore Math

Here are some examples of decimal word problems. We will illustrate how block diagrams can be used to help you to visualize the decimal word problems in terms of the information given and the data that needs to be found. Block diagrams or bar modeling are used in Singapore Math and tape diagrams are used in Common Core Math.

Example: The length of a ribbon is 1.28 m. The length of a rope is 2.74 m longer than the ribbon. What is the length of the rope?

1.28 + 2.74 = 4.02

The length of the rope is 4.02 m.

Example: The mass of a jar of sugar is 1.9 kg. What is the total mass of 4 such jars of sugar?

1.9 × 4 = 7.6

The total mass of 4 such jars of sugar is 7.6 kg.

Example: A pail holds 5.2 l of water. A bottle holds 3.9 l less water than the pail. What is the volume of water in the bottle?

5.2 – 3.9 = 1.3

The volume of water in the bottle is 1.3 l.

Example: Susan has 4 times as much money as her sister. If Susan has $10, how much money does her sister have?

$10 ÷ 4 = $2.50

Susan’s sister has $2.50.

Decimal Word Problems: Addition, Subtraction, Multiplication, Division

• Maneesha purchased a box of pencils for $1.28 and gave the cashier $10.00. How much change should she get back?
• If you buy an ebook for $29.62 and download 5 songs for $1.29 each, what is the total amount you have spent?
• Emilio’s batting average in his first year playing baseball was 0.089. In his second year, he improved to an average of 0.29. His third year, he improved even more to an average of 0.329. What is Emilio’s average over the three years? What is the difference between the first and third year averages?
• Shanelle purchased 4 pencils for $0.28 each. If she had a $5 bill, how much money did she have left after purchasing the pencils?
• A train took 1.2 hours to go 73.8 miles from Cary to Fayetteville. Find the rate of the train.
• I have a pile of DVD’s. Each DVD has a height of 0.3 cm. If the pile is 75 cm tall, how many DVD’s are there in the pile?

Word Problems With Decimals

Solve word problems involving addition, subtraction, multiplication and division of decimal numbers.

• Matt deposits a check for $234.95 into his checking account. He now has a total of $1,479.87 in the account. How much was the account before the deposit?
• Stan compares his checkbook record with his monthly bank statement that says he has $876.47. Stan sees that checks for $32.85, $97.10 and $158.78 have not been cashed yet. How much money does Stan really have available?
• An ad for a computer system lists the price as $899.95. There is an instant rebate of $55.55 and a mail-in rebate of $66.66. What is the final price of the system after both rebates?
• At work, Amy receives $22.25 per hour for up to 40 hours per week. Any time beyond that is paid at a rate of $37.80 per hour. If she receives $1,173.50 in her paycheck, how much time did she work that week?
• The total receipts for a basketball game is $1,400 for 788 tickets sold. Adults pay $2.50 and students pay $1.25. How many tickets of each kind were sold?

Decimals Word Problem Using Block Model

Solving a 2-part decimals word problem using block modeling.

Example: David took a walk around a park twice. He took 12.4 minutes to walk the first round. In the second round, he took 3.2 fewer minutes than he did the first round. How long did David take to complete his walk altogether?

How to solve decimal word problems using tape diagrams?

The following video shows an example of a decimal word problem.

Example: Manny tracked the amount of food he ate from Monday to Friday. He ate 16.3 pounds of food. On Monday, he ate 3.2 pounds and on Tuesday, he ate 2.9 pounds. He ate an equal amount on the other three days. How much did he eat on those days?

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Worksheet on Word Problems on Addition and Subtraction of Decimals

Practice the questions given in the worksheet on word problems on addition and subtraction of decimals. Read the questions carefully to add or subtract the decimals as required.

1. Tania bought a book for $152.75, a pen for $45.25 and a chocolate for $28.75. What amount did she spend?

2. Nancy bought biscuits for $51.25. She gave a $100 note to the shopkeeper. How much dis she get back from the shopkeeper?

3. Mary had $305.80 in her bank account. She deposited $250.25 more and then withdrew $317.50 from her account. What is the balance now in her account?

4. Mike wants to buy a Physics book costing $600. He has $475.25 only in his purse. How much more money does he need to purchase the book?

5. Ron purchased a bag for $134.60, a book for $328.23 and a tie for $80.55. How much is left with him if he had $600 in all?

6. The difference of two decimals are 68.09. The smaller one is 353.48. Find the other one.

7. The sum of three decimals are 938.629. Two of them are 456.54 and 392.69. Find the third one.

8. Rachel had $739.68. She gave $235.09 to Jessica, $345.45 to Rebecca and the remaining money to Sara. How much did she give to Sara?

9. Jaclyn weighs 27.14 kg, Mary weighs 31.37 kg and Jenny weighs 28.38 kg. What is their total weight?

10. Kate travelled 320.25 km and Maya travelled 236.38 km. Who travelled more and by what distance?

11. Jack has lost $145.50 in a market. He is now left with $95.75. How much did he have?

12. Noor had $350.50. She bought jeans for $264.50 and a shirt for $65.75. How much did she have after pay?

13. Sam bought a pair of shirts for $205.75, a pant for $225.25 and a coat for $1225.20. What was the total cost of all the three items?

14. The sum of two decimals are 138.28. One of them is $68.42. Find the other one.

15. Jenifer had $178.50 with her. She has spent $138.85. How much money does she have now?

Answers for the worksheet on word problems on addition and subtraction of decimals are given below.

8. $ 159.14

9. 86.89 kg

10. Kate 83.87 km

11. $241.25

13. $1656.50

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Word Problems on Addition and Subtraction of Decimals | Adding and Subtracting Decimals Word Problems

See your kids excel in mathematics taking the help of the free and extensive problems available on decimal addition and subtraction. Use the interactive exercise Word Problems on Addition and Subtraction and develop personalized learning among your kids. This Worksheet on Adding and Subtracting Decimals has an extensive collection of frequently asked problems in your exams. Assess your strengths and weaknesses using the problems over here regarding decimal addition and subtraction and get a good grip on the concept.

Do Read Similar Articles:

• Worksheet on Concept of Decimal
• Simplify Decimals Involving Addition and Subtraction Decimals
• Adding Decimals
• Subtracting Decimals

Word Problems Involving Addition and Subtraction of Decimals

Example 1. There are 8.50 liters of milk in the pan. Raju added 5 g of sugar and 1.25 liters of water to the pan. Find how many liters of milk and water are there in the pan? Solution: No. of liters of milk in the pan = 8.50 No.of liters of water in the pan = 1.25 No. of liters of milk and water in the pan is 8.50 + 1.25 = 9.75 Hence, There are 9.75 liters of milk and water are there in the pan.

Example 2. Raju has 14.50 acres of agricultural land. He decided to give his first son Sai 6.50 acres of land and the second son Sudheer 5.50 acres of land. Find how much land does Raju has after distributing his sons? Solution: Raju has the agricultural land = 14.50 Raju gave the land for the first son = 6.50 Raju gave the land for the second son = 5.50 No. of acres of land Raju distributed for his sons = 6.50 + 5.50 =12.00 No. of acres of land Raju has after distributing his sons =14.50-12.00 = 2.50 Therefore, Raju has 2.50 acres of agricultural land after distribution.

Example 3. The price of the sugar last month is Rs 42.50. This month the price of sugar is increased by Rs 2.50. Find out what is the price of the sugar this month? Solution: The price of the sugar last month = 42.50 The price of the sugar this month is increased by =  2.50 The price of the sugar this month = 42.50 + 2.50 = 45.00 Hence, the price of the sugar this month is Rs 45.

Example 4. Karthik wants to go to the temple which is 150.50 km. Karthik stops driving the car after driving 65.80 km, because of the traffic jam. How much distance he has to travel for going to the temple? Solution: Karthik wants to go to the temple at a distance = 150.50 Karthik traveled by car up to the distance = 65.80 The distance Karthik has to travel for going to the temple = 150.50-65.80 = 84.70 Therefore, Karthik has to travel 84.70 km for going to the temple.

Example 5. Praveen wants to buy a house in Banglore. He went to choose the houses. In the first house, the rooms were 28.50 square feet longer. The second house was 2.7 square feet shorter. The third house was 5.6 square feet longer than the first house. What is the difference in feet between the second and third house rooms? Solution:  The size of the rooms in the first house was = 28.50 The size of the rooms in the second house was shorter by = 2.7 The size of the rooms in the second house was = 28.50 – 2.7 = 25.8 The size of the rooms in the third house was longer than the first house by = 5.6 The size of the rooms in the third house was = 28.50 +5.6 = 34.1 The difference in feet for the second and third house was = 34.1-25.8 = 8.3 Hence, The difference in feet for the second and third houses was 8.3 square feet.

Example 6. Varsha had money Rs 750.80. She bought a dress for Rs 530.20. How much money left with Varsha? Solution: Varsha had money = 750.80 She bought a dress = 530.20 Money left with Varsha = 750.80-530.20 = 220.60 Hence, Money left with Varsha = 220.60

Example 7. Pavan went to a store. He bought 2.25 kg of cashews and almonds. If pavan bought 1.25 kg of almonds, how many kg of cashews? Solution: Total no. of kg of cashews and almonds = 2.25 kg Pavan bought almonds = 1.25 kg No. of kg of cashews = 2.25-1.25 = 1 kg Hence, the total no. of kg of cashews is 1 kg.

Example 8. Sindhu purchased a book for Rs 50.50, a pen for Rs 25.50. How much amount did Sindhu spend? Solution: Sindhu purchased a book = 50.50 Sindhu purchased a pen = 25.50 The amount Sindhu spend = 50.50 +25.50 = 76 Hence, the amount Sindhu spends is Rs 76.

Example 9. Harish has some money. He bought a gift for his friend in the amount of Rs 500.50. Harish is left with the amount of Rs 300. Find the amount of money Harish has before spending the money? Solution: Harish bought a gift for his friend = 500.50 Harish has left with the money = 300 The amount of money Harish has before spending the money = 500.50+300 = 800.50 Therefore, Harish has 800.50 before spending the money.

Example 10. In a juice shop, there are 10.25 liters of orange juice and 12.50 liters of grape juice. How many liters of juices are needed to fill an order of 30 liters of juice? Solution: No. of liters of Orange juice = 10.25 No.of liters of grape juice = 12.50 Total no. of liters of juices in the shop = 10.25+12.50 = 22.75 No. of liters of juices required to fill an order of 30 liters of juice = 30-22.75 = 7.25 Hence, no. of liters of juice required to fill an order is 7.25 liters.

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Real-life problems, working with decimals

Common Core Standards: Grade 4 Measurement & Data , Grade 4 Number & Operations in Base Ten , Grade 5 Number & Operations in Base Ten

CCSS.Math.Content.4.MD.A.2, CCSS.Math.Content.4.NBT.B.5, CCSS.Math.Content.5.NBT.B.7

This worksheet originally published in Math Made Easy for 5th Grade by © Dorling Kindersley Limited .

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Real-life problems, working with decimals #2, real-life problems: money, adding decimals, 5th grade, subtracting decimals.

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Mixed decimals word problems

Add, subtract and multiply decimals.

These grade 5 math word problems involve the addition, subtraction and multiplication of decimal numbers with one or two decimal digits . Some problems may have more than 2 terms, include superfluous data or require the conversion of fractions with denominators of 10 or 100.

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Solving Problems that Include Fractions and Decimals

Introduction.

There are four important operations that you will encounter when solving problems in mathematics. The figures below indicate some of the actions in a problem that lead to different operations.

Addition and subtraction are related operations. Addition typically means to combine two or more numbers, and subtraction involves the difference , or removal, of one number from another.

Multiplication and division are also related operations. Both operations involve grouping and rates.

You have explored how to tell when to use which operation. Now, you will focus on identifying the operation from a word problem, and then use procedures to actually perform the operation and determine a solution to the problem.

Working with Signed Numbers

Signed numbers include integers and other rational numbers that have either a positive or a negative sign.

Source: Badwater Elevation Sign, Complex01 and Elevation Benchmark, Jeff Kramer, Wikimedia Commons

Use the diagram below to review standard procedures for adding, subtracting, multiplying, and dividing integers.

Adding and Subtracting Decimals

You have applied the rules of integers to solve word problems. Now, you will review ways to add and subtract decimals, and then use what you learn to solve problems relating to addition and subtraction of positive and negative decimals.

Click on the image below to open a base-ten model interactive in a new web browser tab or window. The interactive represents the two addends in an addition problem, or the minuend  and subtrahend in a subtraction problem. Use the manipulative to work through at least 3 problems.

• Click on a block and drag it on top of its opposite block to remove zero pairs.
• Click on a block and drag it to the next column to regroup.
• Click “Next Problem” to move to the next problem when you are ready.

Need additional help for addition?

Need additional help for subtraction?

Use what you noticed in the interactive to answer the following questions.

In the original problem, 4.3 – 1.5, when you dragged a ones rod into the tenths column, it split into 10 tenths. How does that relate to the regrouping that was recorded symbolically in the image shown below?

In an addition problem, such as 6.4 + 4.8, when you regroup 10 tenths into 1 one and drag the ones rod into the ones place, how did that action appear in the regrouping that was recorded symbolically such as the regrouping shown in the image below?

Pause and Reflect

1. Why is it important to line up the decimal point when adding or subtracting decimal numbers?

2. When regrouping 1 one and 3 tenths into 13 tenths, why do you cross out the original 3 in the tenths place and replace it with 13? 

Adding and Subtracting Fractions

You have used models and algorithms to add and subtract decimals, paying special attention to the regrouping that was necessary to perform the computations. Now, you will extend the idea of regrouping to models and procedures used to add and subtract fractions, including mixed numbers.

Consider the following problem.

The example below shows how Marley used fraction strips to solve this problem.

Click the image below to view additional examples, including a video with a worked-out example for you to follow.

1. How is regrouping when subtracting mixed numbers similar to regrouping when subtracting decimals?

2. When adding decimals, you regroup when the sum of the two digits in a place value that is greater than 10. When would you need to regroup as you add mixed numbers?

Multiplying and Dividing Decimals

Now that you’ve investigated addition and subtraction with decimals and fractions, let’s take a closer look at multiplication and division. You will start in this section with decimals, and then use a similar model to multiply and divide fractions and mixed numbers in the next section.

• Write an expression that you can use to determine the amount of oil that Rachel started with.
• How would you represent 2.2 and 2.5 as improper fractions with denominators of 10?

The interactive below uses blocks to multiply decimals. When the blocks are combined, they will form a rectangle; the area of the rectangle is the product of the two decimals or the answer to Rachel’s problem.

• In the first activity, the first decimal is the length of the rectangle, and the second decimal is the width. Represent each decimal by dragging the appropriate blocks and moving them to the area for each decimal.
• In the second activity, use the information from the decimals and drag the blocks to the open area to create a rectangle. You will use the green blocks to fill in the missing pieces of the rectangle.
• Is the answer the same as what we found earlier in Anu's solution?
• Adjust the numerators to create and represent two more multiplication problems. Record those problems on a piece of paper.

Based on what you saw in the interactive, why do you think that the product has the same number of digits to the right of the decimal as the total number of digits to the right of the decimal in the two factors ?

Multiplying and Dividing Fractions

In this section, you will look at models to represent multiplying and dividing fractions.

Multiplying Fractions

Use the interactive below to represent the problem and graphically illustrate the product. Use the Numerator and Denominator sliders to create each fraction or mixed number. You may also need to use the Zoom in/out sliders to see the entire model.

Need additional directions?

Use the interactive to answer the following questions:

• What are the dimensions of the shaded rectangle in the solution? Check Your Answer
• The solid lines represent the boundaries of a rectangle with an area of 1 square unit. The dashed lines represent the boundaries of a number of equal-sized regions within this area. What fraction of 1 does each smaller rectangle represent? Check Your Answer

• What mixed number does this rearranged figure represent? How does this compare with the product of 3 4 and 6 1 2 ? Check Your Answer

Dividing Fractions

To solve this problem, Barbara used a fraction strip generator, which gave her the following diagram.

• Barbara knew this was a division problem, not a multiplication problem. How did she know that? Check Your Answer
• Use the diagram to explain why the quotient of 6 1 2 ÷ 1 2 is 13. Check Your Answer

Use the same fraction strip generator that Barbara used to solve the problem below.

Click the image below to open the fraction strip generator in a new web browser tab or window. Enter the key information from the problem, including the dividend and the divisor , and then use the results to answer the questions that follow.

In the fraction diagrams, both 5 3 4 and 3 8 are marked off into eighths. Why do you think that is the case? Check Your Answer

To divide 5 3 4 by 3 8 , the number sentence beneath the diagrams shows multiplication of 5 3 4 by 8 3 , which is the reciprocal of 3 8 . Multiplying by 8 3 is the same as multiplying by 8 , and then dividing by 3 . Why do you need to multiply 5 3 4 by 8 , which is the numerator of the reciprocal? Check Your Answer

The next step in the number sentence divides the product of 5 3 4 and 8 by 3 (multiplies 5 3 4 by the fraction 8 3 ) . Why do you need to divide by 3 at this point? Check Your Answer

See the completed fraction diagram for Patrice's ornament problem.

Completed fraction diagram

1. How does the multiplication algorithm connect to the area model that you used in the first interactive?

2. How does the division algorithm connect to the fraction strip model that you used in the interactive?

You studied models that represent operations on rational numbers (fractions and decimals). You also connected those models to the standard algorithms for performing the operations.

The graphic below summarizes procedures to add, subtract, multiply, and divide decimals.

The graphic below summarizes procedures to add, subtract, multiply, or divide fractions, including mixed numbers.

Copy and paste the link code above.

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