One Quantity as a Fraction, Decimal or Percentage of Another

Writing one quantity as a fraction of another starts with a single idea: the part goes on top and the whole goes underneath. Twelve seats out of twenty are taken, so the fraction taken is twelve over twenty. The hard bit is spotting which number is the whole, so listen to the words. Out of twenty, as a fraction of twenty, a test marked out of twenty: in every case twenty is the whole. Watch the classic trap. Eight seats are empty, but the fraction taken is not twelve over eight. Eight is the leftover, not the total. Ask yourself how many there were altogether before anything happened, and put that number on the bottom.

Once you have built the fraction, tidy it up. Twelve over twenty can be simplified by dividing top and bottom by a common factor. The factors of twelve are one, two, three, four, six and twelve; the factors of twenty are one, two, four, five, ten and twenty; so the highest they share is four. Twelve divided by four is three and twenty divided by four is five, giving three fifths. Look at the two bars: the shading stops in exactly the same place, because simplifying changes the names of the numbers, not the amount. The order matters though. Build the fraction from part over whole first, and only then simplify, so you never lose track of the total.

The line in a fraction is really a divide sign, so a fraction is a division waiting to happen. Three fifths means three divided by five, which is nought point six. Seven over twenty-five means seven divided by twenty-five, which is nought point two eight. Three quarters is nought point seven five, and five eighths is nought point six two five. The order matters enormously: always divide the top by the bottom, never the other way round. Here is a free check. Because you are describing a part of a whole, your decimal should land between zero and one. If you get something bigger than one, you have almost certainly divided the wrong way round.

Percent simply means out of one hundred, so a percentage tells you how many parts you would have if the whole were cut into a hundred. Sixty of these hundred squares are shaded, which is sixty percent. To turn a decimal into a percentage, multiply by a hundred and write the percent sign. Nought point six becomes sixty percent, nought point two eight becomes twenty-eight percent, and nought point seven five becomes seventy-five percent. So the full journey runs from twelve out of twenty, to twelve twentieths, to three fifths, to nought point six, to sixty percent — five ways of saying one thing. And percentages do not have to be whole numbers: five eighths is sixty-two point five percent, and that is a perfectly good answer.

One last trap, and it catches almost everybody. When the two quantities have different units, you must convert before you build the fraction. Forty-five minutes as a percentage of two hours is not forty-five over two. Two hours is a hundred and twenty minutes, so the fraction is forty-five over a hundred and twenty. Simplify by dividing both by fifteen to get three eighths, divide three by eight to get nought point three seven five, and multiply by a hundred for thirty-seven point five percent. The same warning applies to thirty centimetres out of two metres, or two hundred and fifty grams out of one kilogram. Match the units first, and the rest is the routine you already know.