Fractions of Quantities

A quarter of twenty means twenty shared into four equal groups, and one of those groups: twenty divided by four, five. That is all a unit fraction of an amount is — a division by the bottom number. A fifth of forty dollars is eight dollars; a tenth of two metres is twenty centimetres, once the two metres is written as two hundred centimetres so the division comes out whole. A quarter of an hour is fifteen minutes for the same reason: sixty divided by four.

For any fraction, find the unit fraction first and then multiply by the top. Three quarters of twenty: a quarter is five, so three quarters are three fives, fifteen. Two fifths of forty dollars: a fifth is eight, two fifths are sixteen. Three tenths of two metres: a tenth is twenty centimetres, three tenths are sixty. Divide by the bottom, multiply by the top, always in that order, because the division is the one that has to come out whole.

When the amount is a picture, the amount is the number of pieces in it. A wall with ten sections: two fifths of ten is four sections, a half is five, so one section is green. A strip of twelve cells, two thirds of it: a third is four cells, two thirds are eight. You can think of it as renaming — two fifths is four tenths — or as fraction-of-an-amount with the amount being ten. Both roads reach four.

The reverse question gives you the part and asks for the whole. A quarter of a number is twelve: forwards was divide by four, so backwards is multiply by four, forty-eight. When the fraction is not a unit fraction, find the unit fraction first. Three fifths of a number is twenty-four: three fifths are twenty-four, so one fifth is eight, and the whole is five eights, forty. Draw a strip with five cells, shade three, share the twenty-four between the shaded cells, and the whole strip is five times one cell. Then check forwards.

In a story, the two questions are which amount the fraction is of, and whether you want the part, the rest, or the whole. Mia spends two fifths of forty dollars: she spends sixteen and has twenty-four left, which you can find either by subtracting or by taking three fifths directly. Turn it round — she spends two fifths and has twenty-four left, what did she have — and twenty-four is three fifths, so a fifth is eight and she had forty. A quarter cut off a three-metre plank leaves three quarters of three hundred centimetres, two hundred and twenty-five. Quiz and exercises now.