Dividing Decimals

Dividing by a decimal looks harder than it is, because there's a move that turns every decimal division into a whole-number division. Shift the decimal point in the divisor — the number you're dividing by — until it becomes a whole number, and shift the point in the dividend the same number of places.

So $6.3 \div 0.9$ becomes $63 \div 9 = 7$ . Both points moved one place right, the value of the quotient didn't change, and the problem turned into basic division facts.

Make the divisor a whole number

Look at the divisor and count how many places its decimal point must move right to make it whole. $0.9$ needs one place; $0.25$ needs two. Then move the dividend's point the same number of places in the same direction.

For $7.2 \div 0.9$ : one shift right turns it into $72 \div 9 = 8$ . The shift must match — moving the divisor without moving the dividend changes the answer.

Why shifting both points is fair

Moving both decimal points one place right is the same as multiplying both numbers by $10$ . A division problem is really a fraction — $6.3 \div 0.9 = \dfrac{6.3}{0.9}$ — and multiplying the top and bottom of a fraction by the same number keeps its value: $\dfrac{6.3}{0.9} = \dfrac{63}{9}$ .

That's why the trick isn't a trick at all. You're rewriting the same problem with friendlier numbers, exactly like writing an equivalent fraction.

When only the dividend has a decimal

If the divisor is already whole, like $8.4 \div 4$ , no shifting is needed. Divide as usual, and bring the decimal point in the answer straight up from its spot in the dividend: $8.4 \div 4 = 2.1$ .

If the digits run out before the division finishes, add zeros to the right end of the dividend and keep going — $0.45$ can be treated as $0.450$ or beyond without changing its value.

Worked examples

Example 1: the divisor is already whole

Divide $8.4 \div 4$ .

The divisor

4

is whole — no shifting needed

Divide; the point comes straight up

8.4 \div 4 = 2.1

Check:

2.1 \cdot 4 = 8.4

✓

Answer: $2.1$

Example 2: shift one place

Divide $7.2 \div 0.9$ .

Move both decimal points one place right

72 \div 9

Divide

72 \div 9 = 8

Answer: $8$

Example 3: a quotient less than 1

Divide $0.45 \div 0.5$ .

Move both decimal points one place right

4.5 \div 5

Divide; the point comes straight up

4.5 \div 5 = 0.9

Answer: $0.9$

Try one yourself

Common questions

Which number's decimal point do I move?

Start with the divisor — the number you're dividing by. Move its point right until it's a whole number, then move the dividend's point the same number of places. Both always move together.

Why am I allowed to move the decimal points at all?

Because moving both points one place right multiplies both numbers by $10$ , and a division problem is a fraction: multiplying top and bottom by the same number keeps the value. $\dfrac{6.3}{0.9}$ and $\dfrac{63}{9}$ are the same number.

What if the dividend runs out of digits?

Add zeros to its right end and keep dividing. $3 \div 0.4$ becomes $30 \div 4$ ; since $4$ doesn't divide $30$ evenly, write $30.0$ and continue: $30 \div 4 = 7.5$ .

Why is

6.3 \div 0.9

bigger than

6.3

divided by

9

Dividing asks how many of the divisor fit inside the dividend. Pieces of size $0.9$ are small, so many fit: $7$ of them. Pieces of size $9$ are big, so less than one fits: $0.7$ . Dividing by a number less than $1$ always gives a bigger quotient than the dividend suggests at first glance.

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