Why is a remainder represented as a fraction?

Answer 1

As multiplication is related to addition, division is related to subtraction: when dividing one number (the dividend) by another (the divisor) the division tells you how many times you can subtract the divisor from the dividend before the result passes zero (and becomes negative). For example:

20 ÷ 5 = 4 tells you that you can subtract 5 four times from 20 (to reach zero), but if you subtract any more it will become negative.

If the dividend is not a multiple of the divisor, then after a certain number of subtractions the result will be greater than zero but also less than the divisor - this is the remainder, eg 22 ÷ 5 = 4 remainder 2, since: 22 - 5 = 17 (once); 17 - 5 = 12 (twice); 12 - 5 = 7 (thrice); 7 - 5 = 2 (four times); if we subtract 5 again, the result is negative (2 - 5 = -3). When dealing with whole numbers, this is as far as we can go and we say 22 ÷ 5 = 4 remainder 2.

However, if we allow for fractions, then we could subtract a fraction of 5 and reach (but not pass) zero. So what fraction of 5 is the remainder 2? Answer: 2/5 (two fifths). So we could subtract a further 2/5 of 5 = 2/3 × 5 = 2 from 2 to get to zero. Thus, if we allow for fractions, we can say 22 ÷ 5 = 4 2/5 (or 4.4 as a decimal).

This is true of all divisors: if there is a remainder greater than zero but less than the divisor, then that remainder (as the numerator) over the divisor (as the denominator) is the fraction of the divisor that needs to be subtracted to reach, but not pass, zero. (This fraction can be simplified, for example 14 ÷ 4 = 3 remainder 2, which with fractions is 3 2/4, but 2/4 can be simplified to 1/2 (as they are equivalent fractions), so 14 ÷ 4 = 3 1/2 (or 3.5 as a decimal).

This the remainder can be represented as a fraction (of the divisor).

Answer 2

In what situation? The remainder of the division of a whole number by another whole number is also a whole number.