Inverse numbers and dividing fractions — Mathematics, 11–13
Two numbers are inverses when their product is 1, and dividing by a fraction is multiplying by its inverse.
Numbers whose product is 1
Two numbers are inverses of each other when their product is 1. 2/3 × 3/2 = 6/6 = 1, so 2/3 and 3/2 are inverses: to find the inverse of a fraction, swap numerator and denominator. A natural number is a fraction over 1: 4 = 4/1, and its inverse is 1/4, because 4 × 1/4 = 1. To divide by a fraction, multiply by its inverse: 5 ÷ 1/4 = 5 × 4 = 20.
How many quarters fit?
Dividing asks how many times one amount fits in another. How many quarter-litre bottles can you fill from 5 litres? Each litre gives 4, so 5 × 4 = 20. Dividing by 1/4 is multiplying by 4, the inverse of 1/4. And sharing 1/2 litre between 2 cups gives 1/2 ÷ 2 = 1/2 × 1/2 = 1/4 of a litre, that is 0.25 l each: dividing by 2 is taking half, that is multiplying by 1/2. Check by multiplying: 20 × 1/4 = 5.
Glasses of 3/8 l, and 2/3 ÷ 4
How many glasses of 3/8 l can you fill with 3/2 l of juice (that is 1.5 l)? Step 1: 3/2 ÷ 3/8. Step 2: the inverse of 3/8 is 8/3, because 3/8 × 8/3 = 24/24 = 1. Step 3: 3/2 × 8/3 = 24/6 = 4. Check: 4 × 3/8 = 12/8 = 3/2. So 4 glasses. Dividing by a natural number works the same way: 2/3 ÷ 4 = 2/3 × 1/4 = 2/12 = 1/6.
Turning the wrong fraction upside down
In 3/4 ÷ 2/5, many people invert the first fraction (4/3 × 2/5 = 8/15), or both, or invert nothing and multiply (3/4 × 2/5 = 6/20). Only the divisor is inverted, and the dividend stays as it is: 3/4 × 5/2 = 15/8. You can always check a division by multiplying the answer by the divisor: 15/8 × 2/5 = 30/40 = 3/4, and you are back at the dividend. Try it with 8/15: 8/15 × 2/5 = 16/75, which is not 3/4.
Cutting rope, counting portions
A 3 m rope is cut into pieces of 3/4 m. How many pieces? 3 ÷ 3/4 = 3 × 4/3 = 12/3 = 4 pieces (4 × 3/4 = 3 m). In a recipe, if each portion needs 1/4 of a cup of sugar and you have 3/2 cups, then 3/2 ÷ 1/4 = 3/2 × 4 = 6 portions. Every time you ask “how many of this fit in that?”, you are dividing, and if the piece is a fraction, its inverse does the job.
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