Chapter 7. Taking a fraction of a quantity

Until now, a fraction stated a quantity: three quarters of a strip, five sixths of a pie. It can also act on a number, and that is a different use. Taking a quarter of 12 marbles is doing something to 12: sharing into four, then taking one part. The result, 3, is a number of marbles, not a fraction.

This use is everywhere in daily life: a third of a wage, three quarters of a distance, half of a duration, two fifths of a stock. It also explains the sense of the words per cent, which are nothing other than a fraction with denominator one hundred applied to a quantity.

The gesture takes two steps, in this order: we divide by the denominator to obtain one part, then we multiply by the numerator to take several of them. Those two steps are the direct translation of what a fraction means, and there is nothing to memorise beyond that.

Taking three quarters of 20 is done like this: 20 shared into four gives 5, and three of those parts give 15. We divided first, then multiplied. The same result is obtained by multiplying first: three times 20 is 60, and 60 shared into four gives 15. Both roads arrive at the same place, and we choose the one whose calculations come out exactly.

In practice, we begin by dividing when the division comes out exactly, because the numbers stay small. We begin by multiplying when it does not: to take two thirds of 10, dividing 10 by 3 does not give a whole number, whereas two times 10 is 20, and it then remains to share 20 into three equal parts.

That last sharing is done without leaving what the book has already settled, provided the counting unit is changed. Each unit is worth three thirds, so 20 units are worth sixty thirds. Sharing sixty thirds into three equal parts gives twenty thirds per part, which is written 20/3. A fraction bar therefore also states a sharing to be done: 20/3 is at once twenty thirds and the result of sharing 20 into three equal parts. This third sense of the bar will be found everywhere as soon as a division does not come out exactly.

Everyday language uses the word of in this sense, and it is a reliable signal: three quarters of 20, half of 48, a third of an hour. Every time the word of links a fraction to a quantity, it announces this two-step calculation.

The multiplication of a fraction by a whole number is the same gesture, said the other way round. Three times 2/5 is 2/5 repeated three times, therefore 6/5, which we obtain by multiplying the numerator alone. The denominator does not move, since the size of the parts has not changed: we simply take more of them.

Chapter 3 announced that one of the habits of whole numbers stops being true: multiplying no longer always makes things bigger. This chapter gives the reason for it. Taking three quarters of a number is multiplying it by a quantity smaller than 1, therefore keeping only a part of it. The result is smaller than the starting number, and that is normal.

The general rule can be read with no calculation. If the fraction is smaller than 1, the result is smaller than the starting quantity. If it is worth 1, the result is equal. If it goes beyond 1, the result is larger. A third of 90 is worth 30, which is smaller; four thirds of 90 are worth 120, which is larger.

This quick reading serves as a check on every calculation in this chapter. A result larger than the starting quantity, when the fraction used was smaller than 1, is wrong: there is no need to go back over the detail in order to know it. Apply this check yourself: that is the exercise along the way below, and its answer is at the end of the volume.