Dividing decimals — Mathematics, 11–13
Dividing by 0.1, 0.01 or 0.001 means multiplying by 10, 100 or 1000, and other divisions with decimals turn into divisions of natural numbers.
How many times does it fit?
Dividing 3 by 0.1 asks how many times 0.1 fits in 3. There are 10 in each unit, so 30 times: 3 ÷ 0.1 = 3 × 10 = 30. In the same way, dividing by 0.01 is multiplying by 100, and dividing by 0.001 is multiplying by 1000. A decimal divided by a natural number can use the natural-number algorithm: 7.5 ÷ 5 → 7.5 = 75 × 0.1, 75 ÷ 5 = 15 and 15 × 0.1 = 1.5.
The smaller the divisor, the bigger the quotient
If the dividend stays the same and the divisor becomes 10 times smaller, it fits 10 times as often, so the quotient becomes 10 times bigger: 6 ÷ 1 = 6; 6 ÷ 0.1 = 60; 6 ÷ 0.01 = 600; 6 ÷ 0.001 = 6000. It is the multiplication rule seen in a mirror: dividing by 0.1 and multiplying by 10 give the same, because 0.1 is a tenth of 1.
3 ÷ 0.25 and 2.4 ÷ 0.02
Work out 3 ÷ 0.25. Step 1: multiply dividend and divisor by 100; the quotient does not change: 300 ÷ 25. Step 2: divide the natural numbers: 300 ÷ 25 = 12. Check with the inverse operation: 12 × 0.25 = 3. Same method for 2.4 ÷ 0.02: × 100 gives 240 ÷ 2 = 120, and 120 × 0.02 = 2.4. In words: 12 quarters of a unit fit in 3.
12 ÷ 0.1 is not 1.2
Because 0.1 “has to do with 10”, many people write 12 ÷ 0.1 = 1.2, that is, they divide by 10. But 0.1 is less than 1, and dividing by a number less than 1 gives a result bigger than the dividend. In 12 there are 120 tenths: 12 ÷ 0.1 = 120. Check: 120 × 0.1 = 12. Dividing by 0.1 is the same as multiplying by 10.
How many portions fit?
Dividing by decimals answers “how many fit?”. How many 0.5 m ribbons can you cut from 4.5 m? 4.5 ÷ 0.5 = 45 ÷ 5 = 9 ribbons. How many 0.25 L bottles can you fill with 3 L? 3 ÷ 0.25 = 12. Before calculating, think whether the quotient should be bigger or smaller than the dividend: dividing by less than 1 gives more.
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