Multiply and divide decimals by decimals

Ontario Grade 7 Mathematics

Number

Grade 7 Mathematics • B2.9

Decimals help us describe amounts smaller than one whole. You may already know how to multiply and divide whole numbers, and how tenths and hundredths show parts of a whole. In this lesson, you will use those ideas with decimal numbers. The key is to keep track of place value. An estimate can help you check that your answer makes sense.

What you will learn

Multiply decimals by decimals and place the decimal point correctly.
Divide a decimal by a decimal by making the divisor a whole number.
Estimate to check whether a decimal answer is reasonable.

1. Make sense of decimal products

A product is the answer to a multiplication question. Imagine a small square divided into one hundred equal parts. One side measures four tenths of a unit and the other side measures three tenths of a unit. The overlap covers twelve of the one hundred small parts. So four tenths multiplied by three tenths is twelve hundredths.

This area model helps explain why decimal multiplication can produce a smaller number. Each side is less than one unit, so the small rectangle is only a fraction of one square unit. In symbols, the model shows

0.4 \times 0.3 = 0.12

.

For larger decimals, first multiply as if the numbers were whole numbers. Then count the decimal places in both factors altogether. A factor is one of the numbers being multiplied. Put that many decimal places in the product, counting from the right. If needed, write zeros to hold empty places.

For example, in

2.4 \times 0.35

, there are three decimal places altogether: one in

2.4

and two in

0.35

. The whole-number multiplication is

24 \times 35 = 840

. The product is

0.840

, which is the same value as

0.84

.

a \times b = ab

Multiply the digits as whole numbers first.
Count all decimal places in the factors.
Place the decimal point so the product has that many decimal places.

2. Divide by a decimal

A quotient is the answer to a division question. The divisor is the number you divide by. Dividing by a decimal can be tricky because the divisor is not a whole number.

Make the divisor a whole number by moving its decimal point to the right. Move the decimal point in the dividend the same number of places. The dividend is the number being divided. This keeps the quotient the same because both numbers have been multiplied by the same amount.

For example, to divide

5.76

by

0.8

, move each decimal point one place right. The question becomes

57.6 \div 8

. Both numbers are ten times as large as before, so their quotient is unchanged. Now divide as usual.

Use an estimate to check the result. Since

0.8

is close to

1

, dividing

5.76

by

0.8

should give a number a little larger than

5.76

. A result near

7

is reasonable.

a \div b = (10a) \div (10b)

Make the divisor a whole number.
Move both decimal points the same number of places.
Estimate before or after calculating to check the size of the quotient.

3. Check place value and reasonableness

A decimal point is not a decoration. Its position tells the value of each digit. When you multiply, the total number of decimal places determines where the point goes. When you divide, changing both numbers by the same factor preserves the answer.

Zeros can be useful placeholders. In a product, if the whole-number calculation has fewer digits than the required number of decimal places, write zeros at the front of the answer. For example, the whole-number product

3 \times 4

is

12

. With two decimal places in the factors, the product is

0.12

.

Check your answer in two ways. First, recount decimal places or confirm that both numbers in a division question moved equally. Second, estimate using nearby easy numbers. If the exact answer is far from the estimate, check your calculation and decimal point.

As you work, write each step clearly. Do not move a decimal point in only one number when dividing. Do not decide the product's decimal point by guessing how the answer looks.

Use place value, not guesswork, to place decimal points.
In division, make matching decimal-point moves.
Use an estimate to catch an answer that is much too large or too small.

4. Try the methods

The examples show the full reasoning, from the decimal numbers to a calculation with whole-number digits. Read each check as part of the solution. It helps you decide whether the answer is sensible, not just whether the arithmetic is finished.

Follow the place-value steps in order.
Explain why the decimal-point move is valid.

Worked example

Multiply two decimals

Find

2.4 \times 0.35

.

Count the decimal places
The first factor has one decimal place. The second has two. The product must have three decimal places.
$1 + 2 = 3$
Multiply the digits
Ignore the decimal points for this multiplication. Multiply twenty-four by thirty-five.
$24 \times 35 = 840$
Place the decimal point
Count three places from the right in 840. The leading zero keeps the required places clear.
$2.4 \times 0.35 = 0.840 = 0.84$

Answer:

0.84

Check: Since

0.35

is less than one, the product should be less than

2.4

. The answer

0.84

fits that check.

Worked example

Divide by a decimal

Find

5.76 \div 0.8

.

Make the divisor whole
Move the decimal point in 0.8 one place right. Move the decimal point in 5.76 one place right too. Both numbers are multiplied by ten, so the quotient stays the same.
$5.76 \div 0.8 = 57.6 \div 8$
Divide
Eight goes into 57.6 a total of 7.2 times. Check by multiplying the quotient by the divisor.
$57.6 \div 8 = 7.2$
Check
The check returns the original dividend, so the quotient is correct.
$7.2 \times 0.8 = 5.76$

Answer:

7.2

Check: The estimate is a little above 5.76 because the divisor 0.8 is less than 1. The answer 7.2 is reasonable.

Worked example

Divide when both numbers are decimals

Find

3.15 \div 0.15

.

Make the divisor whole
The divisor has two decimal places. Move both decimal points two places right. In 3.15, the point moves to make 315.
$3.15 \div 0.15 = 315 \div 15$
Divide the whole numbers
Fifteen goes into 315 twenty-one times. This gives the quotient of the original division too, because both numbers were scaled equally.
$315 \div 15 = 21$
Check the quotient
Multiply the quotient by the original divisor. The result matches the original dividend.
$21 \times 0.15 = 3.15$

Answer:

21

Check: A quick estimate is

3 \div 0.15

. Fifteen hundredths fits into three about twenty times, so 21 is reasonable.

Common mistakes and how to avoid them

In multiplication, count decimal places in only one factor.

Correction: Count the places in both factors and add the counts before placing the point.

In division, move the divisor's decimal point but leave the dividend unchanged.

Correction: Move both decimal points the same number of places so the quotient stays the same.

Forget a zero needed at the front of a product.

Correction: Count the required decimal places from the right and use a zero as a placeholder.

Accept an answer without checking its size.

Correction: Estimate with nearby easy numbers and compare the estimate with the answer.

Lesson summary

For decimal multiplication, multiply the digits as whole numbers, then count the decimal places in both factors.
For decimal division, move both decimal points equally until the divisor is a whole number.
Estimate and use multiplication to check division answers.

Check your understanding

Question 1

What is

0.6 \times 0.4

?

$2.4$
$0.24$
$0.024$
correctIndex": 1, "explanation": "Six times four is twenty-four, and the factors have two decimal places altogether, so the product is 0.24."}

Show answer and explanation

0.24

Six times four is twenty-four, and the factors have two decimal places altogether, so the product is

0.24

.

Question 2

What is

4.2 \div 0.7

?

$0.6$
$6$
$60$
correctIndex": 1, "explanation": "Move each decimal point one place right to get 42 divided by 7, which equals 6."}

Show answer and explanation

6

Moving both decimal points one place right gives

42 \div 7

, which equals

6

.

Question 3

What is

1.44 \div 0.12

?

$1.2$
$12$
$120$
correctIndex": 1, "explanation": "Move both decimal points two places right to get 144 divided by 12, which equals 12."}

Show answer and explanation

12

Moving both decimal points two places right gives

144 \div 12 = 12

.

Key terms

Factor: A number being multiplied by another number.
Product: The answer to a multiplication question.
Dividend: The number being divided.
Divisor: The number you divide by.
Quotient: The answer to a division question.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B2.9. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B2.9 · Multiply and divide decimals by decimals

Ontario Grade 7 Mathematics

Grade 7 Mathematics • B2.9

What you will learn

1. Make sense of decimal products

2. Divide by a decimal

3. Check place value and reasonableness

4. Try the methods

Multiply two decimals

Divide by a decimal

Divide when both numbers are decimals

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review