Divide by a Decimal: Make the Divisor a Whole Number | Grade 5 Math | LevelKite

You already know how to divide a decimal by a whole number. In this lesson, the number you divide by is a decimal too. One idea turns this new kind of problem into the kind you already know.

You will need three words. In $0.8 \div 0.2 = 4$ , the dividend is $0.8$ , the number being divided. The divisor is $0.2$ , the number you divide by. The quotient is $4$ , the answer.

What does $0.8 \div 0.2$ ask? It asks how many groups of $0.2$ fit in $0.8$ . Say both numbers in tenths: $0.8$ is 8 tenths, and $0.2$ is 2 tenths. Four groups of 2 tenths make 8 tenths, because $2 + 2 + 2 + 2 = 8$ . So 4 groups fit, and $0.8 \div 0.2 = 4$ .

One question, asked in four place values

Division	In words	Quotient
$0.08 \div 0.02$	How many groups of 2 hundredths fit in 8 hundredths?	$4$
$0.8 \div 0.2$	How many groups of 2 tenths fit in 8 tenths?	$4$
$8 \div 2$	How many groups of 2 ones fit in 8 ones?	$4$
$80 \div 20$	How many groups of 2 tens fit in 8 tens?	$4$

Read down the table. To go from one row to the next, you multiply the dividend by $10$ and the divisor by $10$ . The quotient is $4$ every time.

Here is why. Think of the dividend as the total and the divisor as the size of one group. Start with 4 groups of 2 tenths. They make 8 tenths. Now make each group 10 times as large. The 4 groups then make a total that is 10 times as large: 4 groups of 2 ones make 8 ones. The total and the group size both changed, but the number of groups is still 4.

Numbers such as $10$ , $100$ and $1{,}000$ have a name: powers of ten. You can build each one from 10s alone: $100 = 10 \times 10$ and $1{,}000 = 10 \times 10 \times 10$ . The same thinking works for each of them. Jump two rows, from the hundredths row to the ones row. You multiply both numbers by $100$ , and the quotient is still $4$ :

0.08 \div 0.02 = (0.08 \times 100) \div (0.02 \times 100) = 8 \div 2 = 4

Key concept

Same power of ten, same quotient

When you multiply the dividend and the divisor by the same power of ten, the quotient does not change. Both the total and the size of each group become 10 times as large, or both become 100 times as large, so the same number of groups still fits. The formula shows the idea for $10$ . Any power of ten works, as long as you use the same one for both numbers.

\text{dividend} \div \text{divisor} = (\text{dividend} \times 10) \div (\text{divisor} \times 10)

Now use this idea to divide by a decimal. You already know how to divide when the divisor is a whole number, so turn the problem into one like that.

Look at the divisor. Choose the power of ten that makes it a whole number: $10$ if it has one digit after the decimal point, $100$ if it has two.
Multiply the dividend and the divisor by that same power of ten.
Divide. The quotient of the new division is also the quotient of the division you started with.

Only the divisor has to become a whole number. The dividend may still be a decimal.

Remember what multiplying by a power of ten does. Multiplying by $10$ moves every digit one place to the left, so the decimal point looks as if it has moved one place to the right: $1.84 \times 10 = 18.4$ . Multiplying by $100$ moves every digit two places to the left: $0.06 \times 100 = 6$ . When you use the same power of ten for both numbers, the digits move the same number of places in both.

Worked example

Tenths divided by tenths

Find $3.6 \div 0.4$ .

Say what the division asks. 3.6 is 36 tenths and 0.4 is 4 tenths, so it asks how many groups of 4 tenths fit in 36 tenths.
$3.6 \div 0.4 = 36 \text{ tenths} \div 4 \text{ tenths}$
The divisor 0.4 has one digit after the decimal point, so multiply both numbers by 10. Now the 36 tenths and 4 tenths are 36 ones and 4 ones. The divisor is a whole number, and the quotient is the same because you used the same power of ten for both numbers.
$3.6 \times 10 = 36 \quad \text{and} \quad 0.4 \times 10 = 4$
Divide by the whole number 4. Nine groups of 4 make 36.
$36 \div 4 = 9$

Answer

$3.6 \div 0.4 = 9$ . Nine groups of $0.4$ fit in $3.6$ .

Worked example

A divisor with hundredths

Find $6.5 \div 0.25$ .

Look at the divisor. 0.25 has two digits after the decimal point. Multiplying by 10 would give 2.5, which is still not a whole number, so choose 100. In words, 0.25 is 25 hundredths.
$0.25 \times 100 = 25$
Multiply the dividend by the same 100 so that the quotient does not change. The 6 moves to the hundreds place and the 5 moves to the tens place. The ones place is empty, so write a 0 there. In words, 6.5 is 650 hundredths.
$6.5 \times 100 = 650$
Divide by the whole number 25. Twenty groups of 25 make 500, which leaves 150. Six more groups of 25 make 150. That is 26 groups in all.
$650 \div 25 = 26$

Answer

$6.5 \div 0.25 = 26$ . There are 26 groups of $0.25$ in $6.5$ .

Did you notice something about those two quotients? In $3.6 \div 0.4 = 9$ , the quotient $9$ is greater than the dividend $3.6$ . In $6.5 \div 0.25 = 26$ , the quotient is greater than the dividend too.

This makes sense when you think about groups. A group of $0.4$ is smaller than 1 whole, so more than one group fits in each whole. That is why the number of groups, $9$ , is greater than $3.6$ .

The table keeps the dividend at $6$ and changes the divisor. Watch the quotient as the divisor goes from greater than $1$ to less than $1$ . For each decimal divisor, the table first multiplies both numbers by $10$ .

Dividing 6 by different divisors

Division	The divisor is	Quotient	The quotient is
$6 \div 2$	greater than 1	$3$	less than 6
$6 \div 1.5 = 60 \div 15$	greater than 1	$4$	less than 6
$6 \div 1$	equal to 1	$6$	equal to 6
$6 \div 0.5 = 60 \div 5$	less than 1	$12$	greater than 6
$6 \div 0.2 = 60 \div 2$	less than 1	$30$	greater than 6

Worked example

Find and fix a mistake

Sam solved $2.52 \div 0.7$ . Sam moved the decimal point in $0.7$ to get $7$ , and then divided: $2.52 \div 7 = 0.36$ . Find the mistake and correct it.

Test the size of Sam's answer first. The dividend is greater than 0 and the divisor 0.7 is between 0 and 1, so the quotient must be greater than 2.52. Sam's 0.36 is less than 2.52, so it cannot be right.
$0.36 < 2.52$
Find what went wrong. Changing 0.7 to 7 multiplies the divisor by 10, but Sam left the dividend as 2.52. Each group became 10 times as large while the total stayed the same, so only one tenth as many groups fit. Sam answered a different division.
Correct it by multiplying both numbers by 10. Now the total and the group size change together, so the quotient stays the same. The dividend is still a decimal, and that is fine.
$2.52 \times 10 = 25.2 \quad \text{and} \quad 0.7 \times 10 = 7$
Divide the decimal 25.2 by the whole number 7. Three 7s make 21, which leaves 4.2. That is 42 tenths, and 42 tenths divided by 7 is 6 tenths.
$25.2 \div 7 = 3.6$
Test the size again. 3.6 is greater than 2.52, just as the first step said the quotient must be. A size test can show that an answer is wrong. It cannot show by itself that an answer is right, so the careful steps still matter.
$3.6 > 2.52$

Answer

Sam multiplied only the divisor by $10$ . Multiplying both numbers by $10$ gives $25.2 \div 7$ , so $2.52 \div 0.7 = 3.6$ .

The big idea: when you multiply the dividend and the divisor by the same power of ten, the total and the group size change together, so the quotient stays the same. That lets you turn a decimal divisor such as $0.4$ or $0.25$ into a whole number and then divide the way you already know.

Multiply both numbers, and use the same power of ten for each. When the dividend is greater than 0 and the divisor is between 0 and 1, expect a quotient greater than the dividend.

Next, you will round quotients, solve word problems that divide decimals, and check a quotient by multiplying it by the divisor.