Dividing Decimals: Move the Point, Then Divide

Quick answer

To divide by a decimal, move the decimal point in the divisor until it is a whole number, and move the point in the dividend by exactly the same number of places. Then divide as usual. 7.2 divided by 0.8 becomes 72 divided by 8, which is 9. The shift is allowed because multiplying both numbers by the same power of ten leaves the answer unchanged.

What you'll learn

Divide a decimal by a whole number
Divide by a decimal by shifting both numbers
Explain why shifting both numbers leaves the answer unchanged

Two different situations

Dividing decimals splits into two cases, and only one of them needs any new work.

The divisor is	Do
a whole number	divide as usual, bring the point straight up
a decimal	shift both numbers first, then divide as usual

The second case reduces to the first, so there is really only one method.

Dividing by a whole number

Nothing changes except that the answer’s decimal point sits directly above the one in the number being divided.

8.4 \div 4 = 2.1

The point in $8.4$ and the point in $2.1$ are in the same column. That is all the bookkeeping required.

Dividing by a decimal

A decimal divisor is awkward, so get rid of it. Move its point right until it is a whole number, and move the other number’s point the same number of places.

7.2 \div 0.8 \quad\longrightarrow\quad 72 \div 8 = 9

One place each. Now it is an ordinary division.

Why shifting both is allowed

A division is a fraction:

7.2 \div 0.8 = \frac{7.2}{0.8}

Multiplying the top and the bottom by $10$ gives

\frac{7.2 \times 10}{0.8 \times 10} = \frac{72}{8}

which is equivalent fractions — the same rule that has governed every fraction on this site. Scaling both parts by the same number never changes the value.

So the shift is not a trick for tidying the divisor. It is multiplying by $\tfrac{10}{10}$ , which is $1$ .

Why the answer often grows

$6 \div 0.5 = 12$ , which is bigger than $6$ .

Division asks how many of these fit into that. Halves are small, so a lot of them fit — twelve halves make six.

Divide by	The answer
more than $1$	shrinks
exactly $1$	unchanged
less than $1$	grows

The mirror image of multiplying, and the same quick check: decide which way the answer should move before you calculate.

When it does not come out exactly

Add trailing zeros to the number being divided and keep going. $3 = 3.000$ , so the zeros cost nothing and give the division room to continue.

3 \div 8 = 0.375

Worked examples

Common mistakes

Practice problems

Work out $8.4 \div 2$ .

Answer

$4.2$

Full solution

A whole-number divisor, so divide as usual and bring the point up: $4.2$ .
Work out $6.5 \div 5$ .

Answer

$1.3$

Full solution

$65 \div 5 = 13$ , and the point comes straight up, giving $1.3$ .
Work out $3.6 \div 0.6$ .

Hint

Move both points one place.

Answer

$6$

Full solution

$36 \div 6 = 6$ .
Work out $0.8 \div 0.2$ .

Answer

$4$

Full solution

Moving both points one place gives $8 \div 2 = 4$ .
Work out $5 \div 0.5$ .

Answer

$10$

Full solution

Write $5$ as $5.0$ , then move both points: $50 \div 5 = 10$ .

Ten halves make five, so the count is right.
Work out $2.25 \div 0.15$ .

Answer

$15$

Full solution

Move both points two places: $225 \div 15 = 15$ .
Work out $9 \div 4$ .

Hint

It does not come out exactly.

Answer

$2.25$

Full solution

Write $9$ as $9.00$ and continue: $9 \div 4 = 2.25$ .
Work out $0.144 \div 0.12$ .

Answer

$1.2$

Full solution

Move both points two places: $14.4 \div 12 = 1.2$ .
Ribbon costs $0.40 a metre. How many metres for $7.20?

Answer

$18$ metres

Full solution

$7.20 \div 0.40$ . Moving both points two places gives $720 \div 40 = 18$ .
Mia works out $4.8 \div 0.6$ by moving only the divisor’s point, and gets $0.8$ . Find her error.

Hint

Should dividing by $0.6$ make the answer bigger or smaller?

Answer

She shifted one number and not the other. The answer is $8$ .

Full solution

Dividing by a number below $1$ must make the answer bigger than $4.8$ . Her $0.8$ is smaller, so it fails the check immediately.

She computed $4.8 \div 6$ , which is a different problem. Moving both points one place gives $48 \div 6 = 8$ .

Checking by multiplying back: $8 \times 0.6 = 4.8$ ✓

Frequently asked questions

How do I divide by a decimal?

Move the decimal point in the divisor until it is a whole number, then move the point in the other number by the same number of places. Divide as usual. 7.2 divided by 0.8 becomes 72 divided by 8.

Why is moving both points allowed?

Because it multiplies both numbers by the same power of ten, and a division is a fraction — scaling top and bottom by the same amount leaves its value unchanged.

What if only the number being divided is a decimal?

Then nothing needs moving. Divide as usual and bring the decimal point straight up into the answer, directly above where it sits below.

Why is the answer sometimes bigger than what I started with?

Dividing by a number below 1 asks how many small pieces fit into something, and many do. 6 divided by 0.5 is 12, because twelve halves make six.

What if the division does not come out exactly?

Add trailing zeros to the number being divided and keep going. They change nothing, and they give the division room to continue into the decimal places.

What to learn next

What Is a Percent? Percents, Fractions and DecimalsWhat percent means, why it is out of one hundred, and how to move between a percent, a fraction and a decimal without memorising three separate rules.

Key terms in this lesson

Quotient: The quotient is the answer to a division. In 12 ÷ 4 = 3, the 12 is the dividend, the 4 is the divisor, and the 3 is the quotient.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.5.NBT.B.7Number and Operations in Base TenAdd, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
CCSS.MATH.CONTENT.6.NS.B.3The Number SystemFluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

Frequently asked questions

How do I divide by a decimal?

Why is moving both points allowed?

What if only the number being divided is a decimal?

Why is the answer sometimes bigger than what I started with?

What if the division does not come out exactly?

What to learn next

Key terms in this lesson

Standards alignment

More Decimals lessons