Fraction of a quantity; expressing a number as a fraction of another

Imagine a charity fair. 240 people came through the gate, and at the end of the day the organiser stands up to thank everyone. She says two things. First: five eighths of you paid full price. Then: 150 of the 240 of you paid full price. Those two sentences point in opposite directions. The first starts with a fraction and turns it into a number of people. The second starts with two numbers and turns them into a fraction. Two different skills, hiding behind the one word: fraction. And whether her two sentences even agree is a ten-second check, once you know which direction you are facing.

First direction: finding a fraction of an amount. Five eighths of 240. The bottom of the fraction tells you how many equal shares to cut the whole into. Cut 240 into 8 equal strips, and each strip is 240 divided by 8: that's 30. The top tells you how many of those strips to take. Take 5 strips of 30: that's 150. So five eighths of 240 is 150. Exactly what the organiser said. The two announcements agree.

That is the whole method, and here is how to keep it: the bottom cuts, the top takes. Divide by the bottom to cut the whole into shares. Multiply by the top to take your shares.

Your turn for a small one. Find three quarters of 20. Is it 5, 15, or 60?

Have a think. I'll wait. The answer is 15. The bottom cuts: 20 divided by 4 is 5. The top takes: 3 shares of 5 makes 15. If you got 5, you cut but forgot to take. If you got 60, you took without cutting first.

One warning before we change direction. Stay in whole numbers and fractions all the way through. Do not turn the fraction into a decimal in the middle of your working. Here is why. Try one sixth of 240 the decimal way. One sixth is nought point one six six six, going on forever, so you round it. Say, to nought point one seven. Multiply by 240 and you get 40 point 8. Now the exact way: 240 divided by 6 is exactly 40. Forty whole people. The decimal route drifted. The fraction route landed. Examiners watch this exact mistake happen. This is from an examiner's report on a train question built around one sixth:

The very large majority correctly calculated the number of passengers in first or standard class. A common error was to convert fractions to decimals or percentages and not obtain exact values for the number of passengers.

Exact values. Fractions keep them. Rounded decimals leak them.

Now the other direction: writing one number as a fraction of another. A wildlife park has 18 sheep, 12 goats, 9 dogs and 21 chickens. Write the fraction of the animals that are chickens, in its simplest form. Resist the urge to grab the 21. The first move is to find the whole. The question says fraction of the animals, and that means all of them. 18 plus 12 is 30, plus 9 is 39, plus 21 is 60. Sixty animals in total. And the whole goes on the bottom. Always. The part the question asked about, the 21 chickens, goes on top. Twenty-one sixtieths.

Check you have that. Same park: 9 of the 60 animals are dogs. In the fraction of the animals that are dogs, which number sits on the bottom: 9, or 60?

Which one? Decide before I say it. 60. The whole goes on the bottom, and the part goes on top: nine sixtieths of the animals are dogs. If you put the 9 on the bottom, your fraction claims the dogs are the whole park. They are not.

Back to the chickens. Twenty-one sixtieths is correct, but it is not finished, because the question said simplest form. Cancel it. The top and the bottom travel together: divide both by 3, and twenty-one sixtieths becomes seven twentieths. If cancelling feels rusty, the video on equivalent fractions and simplifying by cancelling has the full method. One more habit. Make sure your fraction answers the question that was asked. Chickens were asked for, not the animals that are not chickens. Right part on top, whole on the bottom, then cancel.

Time for the exam side of this topic: where these marks actually go missing. On an animals question just like ours, the examiner's report says this, word for word:

Others gained 2 marks, mostly for giving their fraction not fully simplified or giving a simplified fraction for the animals that were not chickens.

Two ways to lose a mark with mostly-right working: stopping before you cancel, and building a correct fraction of the wrong thing. Both are fixed by the same three-step check. Whole on the bottom. The asked-for part on top. Then cancel. Last question. A publisher prints a batch of 200 books. Three eighths of the batch are hardback, and hardbacks are packed 6 to a box. How many boxes are needed to pack all the hardbacks? Work it all the way through. The bottom cuts, the top takes. And then think about what the boxes need. Take your time. I'll wait right here. Here it is. The bottom cuts: 200 divided by 8 is 25. The top takes: 3 times 25 is 75 hardbacks. Now the packing: 75 divided by 6 is twelve point five boxes. But nobody can order half a box. Twelve boxes hold only 72 books, and 3 hardbacks are left on the floor. The answer is 13 boxes. The maths answer is twelve point five. The final answer is 13. And the reason is the context, not a rounding rule. Even if the division had come out at twelve point one, you would still need 13 boxes, because rounding down leaves books with nowhere to go. When a question lives in the real world, ask what the situation needs. The examiner's report on a question with this exact shape:

This meant that most lost the M1 for their improper fraction/decimal/mixed number rounded up to the next integer.

M1 is a method mark. Students did the fraction work, reached a number like ours, and never asked what the boxes needed. The maths answer is not always the final answer.

The short version, then, to take with you. Two skills share one word. Finding a fraction of an amount: the bottom cuts, the top takes. Five eighths of 240: cut into 8 shares of 30, take 5, get 150. Stay in fractions and whole numbers, with no decimals mid-working. Writing one number as a fraction of another: the whole goes on the bottom, the asked-for part on top, then cancel. And when the question lives in the real world, answer what the context needs. Boxes round up.

Next in the chapter: converting between fractions, decimals and percentages. The same numbers, moving between their three different costumes.

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