Dividing by a decimal

The idea

When the divisor is a decimal, multiply the divisor and the dividend by the same 10, 100, 1000, … to make the divisor a counting number, then divide using place value.

Pune to Matheran

Ravi went from Pune to Matheran by scooter in 2.5 hours. The distance was 126 km. His average speed is 126 ÷ 2.5. Dividing by 2.5 is awkward. We would like the divisor to be a counting number.

What could we do to 126 ÷ 2.5 so that the answer stays the same?

• Multiply only 2.5 by 10
• Multiply both 126 and 2.5 by 10
• Remove the point from 2.5 and leave 126 as it is

Scale both numbers

Multiply by 10 to make the divisor a counting number. Try both buttons and watch the answer in the green bubble.

Same multiplier on both

A division can be written as a fraction. Multiplying the top and the bottom of a fraction by the same number does not change its value. So multiply the dividend and the divisor by the same 10, 100, 1000 …

Multiplying only the divisor changes the answer: 126 ÷ 25 = 5.04, which is 10 times smaller than 126 ÷ 2.5 = 50.4. After scaling, divide by place value as before.

When the divisor is a decimal, multiply the divisor and the dividend by the same 10, 100, 1000, … to make the divisor a counting number. Then divide using place value.

• Division | Counting-number divisor | Answer
• 126 ÷ 2.5 | 1260 ÷ 25 | 50.4
• 4.68 ÷ 1.3 | 46.8 ÷ 13 | 3.6
• 4.68 ÷ 0.13 | 468 ÷ 13 | 36

Notes

If the divisor is a decimal, multiply the dividend and the divisor by the same 10, 100, 1000, … until the divisor is a counting number. The answer does not change. Then divide by place value.

Check yourself

Find 7.2 ÷ 0.4. Make the divisor a counting number first.

Answer: 18

7.2 ÷ 0.4 = 72 ÷ 4 = 18.

To work out 9.6 ÷ 0.32, which is the smallest of 10, 100 and 1000 that we can multiply both numbers by to make the divisor a counting number?

How many 0.25 kg packs can be made from 6 kg of rice? Find 6 ÷ 0.25.

Answer: 24 packs

6 ÷ 0.25 = 600 ÷ 25 = 24 packs.

Which has the same answer as 8.4 ÷ 0.21?

Find 1.5 ÷ 0.4. Scale first, then divide using place value. Type a decimal.

Answer: 3.75

1.5 ÷ 0.4 = 15 ÷ 4 = 3.75.

• 10. 0.32 × 10 = 3.2 is still a decimal. We need 100.
• 100 — correct. 0.32 × 100 = 32, a counting number. And 9.6 × 100 = 960. So it is 960 ÷ 32.
• 1000. That works too, but it is not the smallest: 0.32 × 100 = 32 is already a counting number.
• 84 ÷ 21. The dividend was multiplied by 10 and the divisor by 100. They must have the same multiplier.
• 840 ÷ 21 — correct. Both numbers were multiplied by 100.
• 8.4 ÷ 21. Only the divisor was multiplied by 100, so the answer is 100 times smaller.