When the divisor is a decimal

Ravi went from Pune to Matheran by scooter in 2.5 hours, covering 126 km. What was his average speed? Speed is distance divided by time, so we need 126 ÷ 2.5 — and now the DIVISOR is a decimal.

Convert the divisor into a fraction: 2.5 = 25/10. Dividing by a fraction means multiplying by its reciprocal, so 126 ÷ 25/10 = 126 × 10/25 = 1260/25.

Now the divisor is a counting number and ordinary long division applies. 1260 ÷ 25 = 50.4. So Ravi's average speed was 50.4 km per hour.

Watch what happened to the numbers. The dividend went from 126 to 1260 — multiplied by 10 — and the divisor from 2.5 to 25 — also multiplied by 10. Both were scaled by the same amount, which is why the answer did not change.

See it again with 4.68 ÷ 1.3. The divisor 1.3 is 13/10, so the division becomes 4.68 × 10/13 = 46.8/13. Both have been multiplied by 10.

And 4.68 ÷ 0.13? The divisor is 13/100, so we get 4.68 × 100/13 = 468/13. Both have been multiplied by 100 this time, because the divisor needed two places.

So the rule is: when the divisor is a decimal, multiply it by 10, 100 or 1000 — whatever makes it a counting number — and multiply the DIVIDEND by the same number. Then divide as usual. Written as a fraction it is obvious why this is allowed: 4.68/0.13 = (4.68 × 100)/(0.13 × 100) = 468/13, and multiplying top and bottom by the same number never changes a fraction's value.