Multiply and divide decimals mentally by powers of ten

Ontario Grade 8 Mathematics

Number

Mental strategies for Ontario Grade 8 expectation B2.3

A decimal shows a number of wholes and parts of a whole. For example,

0.6

means six tenths. When we multiply or divide a decimal by

10

,

100

, or

1{,}000

, its digits take on different place values. We can use this pattern to calculate mentally. We will first connect the pattern to place value, then practise multiplying and dividing.

What you will learn

Recognize powers of ten such as 10, 100, and 1,000.
Multiply decimals mentally by powers of ten using place value.
Divide decimals mentally by powers of ten using place value.
Check whether an answer is reasonable.

1. Use place value to see the pattern

Place value tells us what a digit is worth because of where it sits. In

4.7

, the

4

represents four ones and the

7

represents seven tenths. A tenth is one of ten equal parts of a whole. In

4.73

, the

3

represents three hundredths, or three parts out of one hundred equal parts.

Think of

0.6

as six tenths. Each tenth contains ten hundredths, so six tenths is the same amount as sixty hundredths. When we count in hundredths, that amount is

60

. This explains why

0.6

multiplied by

100

equals

60

.

A power of ten in this lesson is a number such as

10

,

100

, or

1{,}000

. Each is ten times the previous one. The number of zeros shows how many place-value steps are involved: one zero means one step, two zeros mean two steps, and three zeros mean three steps.

People often say that the decimal point shifts. More exactly, the digits shift to new places in the written number. The decimal point is written in the usual position after the ones place. This way of thinking keeps our attention on the digits and their values.

10, 100, 1{,}000

Each place to the left is worth ten times as much as the place to its right.
Multiplying by a power of ten makes a positive number larger.
Dividing by a power of ten makes a positive number smaller.

2. Multiply mentally

When multiplying by

10

, move each digit one place left in the written number. When multiplying by

100

, move each digit two places left. Multiplying by

1{,}000

means three places left. The digits stay in the same order.

For example,

3.4

multiplied by

10

is

34

. The

3

moves from the ones place to the tens place, and the

4

moves from tenths to ones. The result is ten times the starting amount.

Sometimes a place has no digit to fill it. Write a zero as a placeholder. A placeholder zero holds a place open so the other digits keep their correct values. For example,

0.08

multiplied by

10

is

0.8

, and multiplied by

100

it is

8

.

Before accepting an answer, check its size. If you multiply a positive decimal by

10

,

100

, or

1{,}000

, the result should be larger. A result that is smaller suggests that the digits moved in the wrong direction.

3.4\times10=34

For multiplication, move the digits left one place for each zero in the power of ten.
Keep the digits in their original order.
Use a placeholder zero when a place would otherwise be empty.

3. Divide mentally

Division by a power of ten works in the opposite direction. Dividing by

10

moves each digit one place right in the written number. Dividing by

100

moves each digit two places right. Dividing by

1{,}000

moves each digit three places right.

For example, in

48\div10

, the

4

moves from tens to ones, and the

8

moves from ones to tenths. The result is

4.8

. Dividing by

100

takes two place-value steps, so

48\div100

is

0.48

.

A zero may be needed before the first digit of the answer. For example, when

6

is divided by

100

, the

6

moves two places right, from ones to hundredths. A zero holds the tenths place, giving

0.06

.

Division by a power of ten should make a positive number smaller. You can also check a division answer by multiplying it by the same power of ten. The product should match the starting number.

48\div100=0.48

For division, move the digits right one place for each zero in the divisor.
Write a zero before the decimal point when the answer is less than one.
Use multiplication by the same power of ten to check a division answer.

4. Choose the direction and check

Start by looking at the operation. Multiplication means move the digits left; division means move them right. Then count the zeros in the power of ten to find the number of places. For example,

1{,}000

has three zeros, so it involves three place-value steps.

Keep the digits in the same order and use placeholder zeros where needed. Do not simply add zeros to the end of every decimal. Instead, work out what each digit is worth in its new place.

Finally, check the size. Multiplication should make a positive number larger, while division should make it smaller. For division, reversing the calculation by multiplying by the divisor is another useful check.

Try saying the direction and number of places before calculating. For

2.35\times100

, say “two places left.” For

2.35\div100

, say “two places right.” This makes it easier to catch a misplaced decimal point.

2.35\times100=235

Count the zeros to find the number of place-value steps.
Multiplication shifts digits left; division shifts them right.
Check both the size of the result and, when useful, the reverse operation.

Worked example

Multiply by 100

Find

4.26\times100

mentally.

Count the places
The multiplier, $100$ , has two zeros. So the digits move two places left in the written number.
$100=10\times10$
Keep the digits in order
The digits $4$ , $2$ , and $6$ stay in that order. After two place-value steps, they represent four hundreds, two tens, and six ones.
$4.26\times100=426$
Check the answer
The answer is larger than $4.26$ , as expected. Dividing it by $100$ returns the starting number.
$426\div100=4.26$

Answer:

426

Check: The answer is one hundred times the starting number, and the reverse calculation returns

4.26

.

Worked example

Divide by 1,000

Find

73\div1{,}000

mentally.

Count the places
The divisor, $1{,}000$ , has three zeros. For division, move the digits three places right.
$1{,}000=10\times10\times10$
Use placeholders
The digits $7$ and $3$ move three places right. Zeros hold the empty places before the $7$ , giving an answer less than one.
$73\div1{,}000=0.073$
Check by multiplying
Multiplying the answer by $1{,}000$ returns $73$ , so the division is consistent.
$0.073\times1{,}000=73$

Answer:

0.073

Check: The answer is smaller than

73

, as expected when dividing by

1{,}000

.

Worked example

Compare multiplication and division

Find both

6.08\times10

and

6.08\div10

.

Multiply
There is one zero in $10$ , so move the digits one place left. The result should be larger than $6.08$ .
$6.08\times10=60.8$
Divide
For division by $10$ , move the digits one place right. The zero holds the tenths place, keeping the $6$ and $8$ in the correct positions.
$6.08\div10=0.608$
Compare
The multiplication result is larger than the starting number, and the division result is smaller. This matches the expected changes in size.
$60.8>6.08>0.608$

Answer:

60.8

and

0.608

Check: Multiplication increased the number and division decreased it.

Common mistakes and how to avoid them

Moving the digits right when multiplying.

Correction: Multiplication by a power of ten moves digits left. Division moves them right.

Moving the digits the correct way but by the wrong number of places.

Correction: Count the zeros in the power of ten. Each zero means one place-value step.

Leaving out a zero that is needed to hold a place.

Correction: Use a placeholder zero when needed. For example,

6\div100=0.06

.

Adding zeros to the end of every decimal when multiplying.

Correction: Use place value to move the digits. The answer may need a decimal point or a placeholder zero.

Lesson summary

A power of ten such as $10$ , $100$ , or $1{,}000$ has one, two, or three zeros.
For multiplication, move digits left one place for each zero.
For division, move digits right one place for each zero.
Use placeholder zeros when needed, then check the size of the answer.

Check your understanding

Question 1

What is

0.39\times10

?

$0.039$
$3.9$
$39$
correctIndex":1,"explanation":"There is one zero in $10$ , so shift the digits one place left. The answer is $3.9$ ."}

Show answer and explanation

3.9

There is one zero in

10

, so the digits move one place left. The answer is

3.9

.

Question 2

What is

5.2\div100

?

$0.52$
$0.052$
$520$
correctIndex":1,"explanation":"Dividing by $100$ shifts the digits two places right. The answer is $0.052$ ."}

Show answer and explanation

0.052

Dividing by

100

moves the digits two places right. The answer is

0.052

.

Question 3

What is

0.74\times1{,}000

?

$74$
$740$
$0.074$
correctIndex":1,"explanation":"The multiplier has three zeros, so the digits shift three places left. The result is $740$ ."}

Show answer and explanation

740

The multiplier has three zeros, so the digits move three places left. The answer is

740

.

Key terms

Decimal: A number written with a decimal point to show ones and parts of a whole.
Place value: The value of a digit because of its position in a number.
Placeholder zero: A zero that holds an empty place so the other digits keep their correct values.
Power of ten: A number such as $10$ , $100$ , or $1{,}000$ , made by multiplying ten by itself.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B2.3. 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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B2.3 · Multiply and divide decimals mentally by powers of ten

Ontario Grade 8 Mathematics

Mental strategies for Ontario Grade 8 expectation B2.3

What you will learn

1. Use place value to see the pattern

2. Multiply mentally

3. Divide mentally

4. Choose the direction and check

Multiply by 100

Divide by 1,000

Compare multiplication and division

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review