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4. Add, Subtract, and Multiply Decimals
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Pre-Algebra /
Chapter 3
4. 

Add, Subtract, and Multiply Decimals

This lesson offers a comprehensive guide to understanding how to add, subtract, and multiply decimals. It uses everyday scenarios, such as calculating video lengths and estimating payments, to make the concepts relatable and practical. The lesson emphasizes the importance of aligning decimal points correctly when adding or subtracting and counting the number of decimal places when multiplying. It also provides tips on rounding numbers to the greatest place value for easier mental calculations. The aim is to equip you with the skills to handle decimal operations in various contexts, from academic exercises to real-world applications like budgeting or measuring.

Student Learning Objectives:
  • Understand the place value chart
  • Add, subtract, and multiply decimal numbers
  • Estimate the product of decimal numbers
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12 Theory slides
13 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Add, Subtract, and Multiply Decimals
Slide of 12
In this lesson, strategies for adding, subtracting, and multiplying integers will be expanded to include decimal numbers. Although performing such operations on decimals is similar to performing them on integers, there are certain rules that need to be followed.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Challenge

A Rectangular Model

A rectangle can be used to represent the product of two whole numbers. The rectangle has a width of 2 units and a length of 3 units. The area of the rectangle is the product of the two numbers, 6.

What would happen if the side length of each square was 0.1 unit long? Could the rectangle be used to represent 0.2* 0.3? Consider the following questions.

a

What is the area of a single square in the new rectangle?

b

Use the small squares to find the product of 0.2 and 0.3.

Discussion

Place Value Chart

The place value chart is a table that displays the correct place of a digit in a number. Place value charts are useful for ensuring that digits are aligned correctly. For example, consider a decimal number. Decimal Number 673.452 The digits of the number are written in the place value chart below.

Place value chart
 Place value charts can be used when adding and subtracting decimals.
Discussion

Adding and Subtracting Decimals

When adding and subtracting decimals, line up the decimal points before adding adding and subtracting as with integer numbers. Consider the following sum. 3.4 + 12.802 There are three steps to finding this sum.

1
Line up the Decimal Points
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The first step is to line up the decimal points. The sum is written so that the decimal points are on top of each other.

Lining up the numbers 3.4 and 12.802

The numbers are now written vertically. A place value chart can also be used to align the decimal places.

Place value chart

2
Place Zeros at the End of a Decimal If Needed
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The first number does not have the same number of visible decimal places as the second number. Place zeros so that the numbers have the same number of decimal places.

Inserting zeros after the decimal point for the number 3.4

Now that the decimal digits in each of the numbers are equal, it will be easier to add or subtract these numbers.

3
Add Digits with the Same Place Value
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Temporarily ignore the decimal point and add the digits that have the same place value. This is the same as adding two integers. After completing all the calculations, bring down the decimal point.

Addition of 3.400 and 12.802

The sum of the given numbers is 16.202.

Note that the subtraction of decimals is done in the same way.
Example

How Long Is Zosia's Video?

Zosia recorded a video of herself singing.

Girl-video-editing.jpg

She recorded two video clips. The first video clip is 4.56 minutes long and the second is 5.54 minutes long.

a

Zosia merged these two video clips. What is the total length of the video?

b

If Zosia removes 1.27 minutes from the video, what is the final length of the video?

Hint
a

Line up the decimal points so that place-value positions correspond to each other before adding any decimal numbers.

b

Line up the decimal points so that place-value positions correspond to each other before doing the subtraction. If necessary, add zeros so that the numbers have the same number of decimal places.

Solution
a

The total length of the video is the sum of the lengths of the clips. First, we line up the decimal points so that the place-value positions match.

& 4.56 + & 5.54 Now we add the place values from right to left, carrying when necessary. We need to be careful to place the decimal point in the correct place value positions.

Zosia’s video is 10.1 minutes long.

b

The final length of the video will be the difference between the full 10.1 minutes and the 1.27 minutes she removes. The numbers have different numbers of decimal places, so we need to add a zero to the end of the full length.

& 10 .1 0 - & 1 .27 Now we can subtract.

The final length of the video is 8.83 minutes.

Pop Quiz

Finding the Sum and Difference of Decimals

Find the sum or difference shown of the given decimal numbers.

Discussion

Estimating Products by Rounding

Estimation is used to find a value that is close to the exact value of a product. This can be done by rounding the factors in the product to the greatest place value — the first non-zero digit on the left in a number. For example, consider the product of two decimal numbers. 35.92 * 0.014 We can estimate this product in two steps.

1
Round The Factors
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The factors in the product will be rounded to the greatest place value to make it easier to compute mentally. In the first factor 35.92, the digit with the greatest place value is 3. In the other factor 0.014, the digit with the greatest place value is 1. 35.92 * 0.014 Next, look at the digit to the right of the greatest place.

  • If the digit is less than 5, the underlined number remains the same.
  • If the digit is 5 or greater, add 1 to the underlined number.

Next, make all digits to the right of the greatest place value 0. For 35.92, the digit to the right of the greatest place value is 5. For 0.014, the digit to the right of the greatest place value is 4.

Estimate 35. 92 * 0.014
Factor Greatest Place Value Digit to the Right
3 5.92 Tens 5
0.01 4 Hundredths 4

The first factor will be rounded to the nearest ten. Since 5 is greater than or equal to 5, add 1 to the number in the rounded place 3. All place values to the right of 3 are replaced with 0. 35.92 ≈ 40.00 (or 40) Similarly, for 0.014, the digit to the right of the greatest place value is 4. This is less than 5, so the 1 remains the same and all place values to the right of 1 become 0. 0.014 ≈ 0.010 (or 0.01) The numbers have now been rounded to their greatest place values. The procedure is visualized in the following applet.

2
Multiply The Rounded Factors
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Now multiply the rounded numbers. Since 0.01 is 10^(- 2), it will move the decimal point in the other factor two places to the left. cr & 40 * & 0.01 & 0.40 The given product is about 0.4. 35.92 * 0.014 ≈ 0.4
Discussion

Multiplying Decimals

Multiplying decimals is similar to multiplying integers. Multiply the numbers first, then place the decimal point in the product. Consider the example product. 1.69 * 3.7 The following steps can be followed to multiply decimals.

1
Multiply Ignoring the Decimal Point
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Ignore the decimal points for now and multiply the numbers as if they were integers.

Multiplication of 169 by 37

2
Count the Number of Decimal Places in the Factors
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Count the total number of decimal places in each factor.

Showing the Number of Decimal Places

Add these numbers to find the total number of decimal places in the product. The sum of the decimal places for this product is 3.

3
Locate the Decimal Point in the Product
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The number of decimal places in the product is the sum of the decimal places in the factors. There are a total of 3 decimal places in the example factors, so the product will have 3 decimal places as well.

Showing Decimal Point in the Product

The product of 1.69 and 3.7 is 6.253.

Example

How Much Has Zosia Made?

Zosia designs and sells digital sticker packs.

Zosia and a friend on her computer
External credits: upklyak

She earns $1.75 for each sticker pack she sells. So far, 59 people purchased a sticker pack.

a

Estimate how much money Zosia has made.

b

Find the exact amount of money that Zosia has made.

Hint
a

Round the numbers to the greatest place value to make it easier to calculate mentally. Round 59 to the nearest ten and 1.75 to the nearest whole number.

b

Multiply decimal numbers like multiplying whole numbers. Place the decimal point in the product by counting the number of decimal places in each factor.

Solution
a

We can estimate the amount of money Zosia has earned by rounding the given numbers. Let's round 59 to the nearest ten and 1.75 to the nearest whole number.

Round Factors 59 * 1.75 ↓ ↓ 60 * 2 The product of 60 and 2 is 120, so Zosia has earned about $120 from her sticker packs.

b

To find the exact amount Zosia has made, we can multiply the number of packs she sold by the price of each pack.

59 * 1.75 Let's start by finding the product as if both numbers were whole numbers.

The product of 59 and 175 is 10 325. Since there are two decimal places in 1.75, the product of 59 and 1.75 must be a decimal number with two decimal places. r 59 * 1. 75 103. 25 l→ → ← r 0 decimal places + 2 decimal places 2 decimal places Zosia has earned $103.25 from her digital sticker packs.

Example

How Much Does The Platform Pay?

Zosia posted a video of herself singing on a video sharing platform. She earns money each time someone watches it.

Girl-and-boy-video-editing.jpg

This week, the platform paid her 0.113 times the money she earned for selling digital sticker packs.

a

Recall that Zosia made $103.25 on selling sticker packs. Estimate the amount that the she earned on the video sharing platform this week by rounding the factors to their greatest place values.

b

Find the exact amount that she made from the video sharing platform.

Hint
a

Round the numbers to the greatest place value to make it easier to calculate mentally. In this case, round 103.25 to the nearest hundred and 0.113 to the nearest tenth.

b

Multiply as though the decimals were whole numbers. Place the decimal point in the product by counting the number of decimal places in each factor.

Solution
a

This week, Zosia earned $103.25 * 0.113 from her video. Let's round the factors to the greatest place value to find an estimate for the product.

Round Factors 103.25 * 0.1 13 ↓ ↓ 100 * 0.1 Multiplying a number by 0.1 means dividing the number by 10 because 0.1 = 110. This means that the product of 100 and 0.1 is 10.

100 * 0.1
Rewrite

Rewrite 0.1 as 1/10

100 * 1/10
MoveLeftFacToNumOne

a* 1/b= a/b

100/10
CalcQuot

Calculate quotient

10

The platform pays about $10 for 59 views.

b

To multiply 103.25 and 0.113, let's first multiply them as if the factors were whole numbers. That is, we start by multiplying 10 325 and 113.

Multiplication of the numbers step by step

There are 2 decimal places in 103.25 and 3 decimal places in 0.113. Their product must be a decimal number with 2+3=5 decimal places.

Placing the Decimal Point Step by Step

The product of the numbers is 11.66725. Since money is represented by numbers with two decimal places, let's round 11.66275 to two decimal places. 11.66275 & Round ⟶ & 11.67 Zosia earned $11.67 from views of her singing video last week. This is close to the estimate we found in Part A, so this answer is reasonable.

Pop Quiz

Finding the Product of Decimals

Multiply the decimal numbers. Make sure that the decimal point is placed correctly!

Closure

Modeling the Product of Numbers

Adding, subtracting, and multiplying decimal numbers works much like it does with whole numbers. The key is placing the decimal point correctly. Let's take another look at the rectangle model for multiplying two numbers.

a

What is the area of a single square in the new rectangle if the side length is 0.1 unit?

b

Use the small squares to find the product of 0.2 and 0.3.

Hint
a

The area of a square is the product of its two sides.

b

Use the area of the squares to find the area of the rectangle.

Solution
a

The given rectangle consists of six squares. Each square has a side length of 0.1 unit.

Rectangle Divided into 6 Squares and One of them is Colored

Since the area of a square is the product of its two sides, this square has an area of 0.1* 0.1 square units. Let's multiply the decimals as if they were whole numbers. 1 * 1 =1 Then we count the number of decimal places in each factor and add them. rr 0. 1 * 0. 1 & l→ → & r 1 decimal place + 1 decimal place 2 decimal places The number of decimal places in the product must be 2. Let's add zeros to the left of 1 and move the decimal point two places to the left. rr 0. 1 * 0. 1 0. 01 & l→ → ← & r 1 decimal place + 1 decimal place 2 decimal places The area of a single square is 0.01 square unit.

b

The area of the rectangle is the product of 0.2 and 0.3. This means that finding the area of the rectangle is the same as finding 0.2 * 0.3.

Rectangle with Side Length

Since there are 6 small squares of area 0.01 each, we can find the area by multiplying 0.01 by 6. First, let's multiply them as if they were whole numbers. 1 * 6 = 6 The sum of the number of decimal places in each factor is 2, so the product will have 2 decimal places. rr 0. 01 * 6 0. 06 & l→ → ← & rr & 2 decimal places + & 0 decimal places & 2 decimal places The area of the rectangle is 0.06 square units. This is also equal to the product of 0.2 and 0.3.



Level 1
Level 2
Level 3
Add, Subtract, and Multiply Decimals
Exercise 3.1

Complete the sentence with the option that makes the given sentence always true.

The sum of two decimal numbers is a whole number when .

The sum of two decimal numbers is equal to a whole number if the decimal parts of the numbers add up to 1. For example, let's add 2.6 and 3.4. The decimal parts of the numbers are 0.6 and 0.4, and the sum of these decimals is 1. cl & 1 & 2 . 6 + & 3 . 4 & 6 . 0 We get 6, which is a whole number. Let's check another example sum with where each addend has two decimal places Consider, for example, the sum of 2.45 and 6.55. cl & 1 1 & 2 . 4 5 + & 6 . 5 5 & 9 . 0 0 The sum of the numbers is a whole number and the sum of the decimal parts is equal to 1. The answer is D.

The sum of two decimal numbers is a whole number when the sum of their decimal parts is 1.


Let's see why the other options do not always make the statement true.

Option A

When the sum of the decimal parts of two numbers is 0.5, the sum of the numbers does have a decimal part. For example, let's add 1.25 and 4.25. cl & 1 . 2 5 + & 4 . 2 5 & 5 . 5 0 This sum is not a whole number.

Option B and Option C

If the decimal parts of two decimal numbers are the same, their sum might be a whole number. However, this is not always the case — let's examine the following sums. & cl & 1 & 6 . 5 + & 2 . 5 & 9 . 0 & cl & & 6 . 4 + & 2 . 4 & 8 . 8 & 0.3cm ⇓ & 1cm ⇓ Sum: & Whole Number & 0.8cm Decimal Number Note that these sums can also be regarded as examples for Option C because the difference between the decimal parts of the numbers is 0.

E
Hint & Answer
S
Solution

Determine if the statement is always, sometimes, or never true.

The product of two decimals, each less than 1, is less than either of the factors.

We want to determine if the given statement is always, sometimes, or never true.

The product of two decimals, each less than 1, is less than either of the factors.

When we multiply a whole number by a number less than 1, the result will always be less than the multiplicand. This is because the product represents some part of the multiplicand. For example, the product of 53 and 0.1 is 5.3. This number is less than 53. |ccccc| Multiplicand & & Multiplier & & Product 53 & *& 0.1& = & 5.3 ⇓ 5.3 < 53 Similarly, when we multiply two decimal numbers, both less than 1, the result will always be less than either of the factors. This is because we are finding a part of a part. Let's consider an example product. 0.4 * 0.7 = 0.28 Here, the product 0.28 is less than both of the decimal factors. This is because we are taking four-tenths of 0.7, or seven-tenths of 0.4.

Note that the red parts on each number line are equal and represent 0.28. They are also smaller than both blue parts. Therefore, when we multiply two decimal factors where both are less than 1, the product will be less than both. We can safely say that the statement is always true.

E
Hint & Answer
S
Solution

Add, Subtract, and Multiply Decimals

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