Comparing two fractions by giving them a common fractional unit

0:00Here are two groups of children, and each group is about to share some chocolate bars. In the first group, one bar between two children. In the second group, five bars among eight children. Which group of children does better? It is tempting to say the first. Two children sharing is surely better than eight. But that reasoning only counts the children. The second group has more food as well as more mouths, and those two pull in opposite directions.

0:33So the picture will not settle it, and neither will staring at the two fractions. One half and five eighths are counts of different things. Here is the move that settles it. One half and five eighths cannot be ranked as they stand, because a half and an eighth are different sized steps. It is like being handed one distance in miles and another in kilometres. Before you can say which is longer, you have to put them in the same unit.

1:07So rewrite one of them. One half is the same length as four eighths — fold the half in half twice, and the shading has not moved at all. Now both fractions count eighths. Four eighths against five eighths. And that is a question about four and five. Five is bigger, so the second group does better after all. More children, but more food than the extra children cost. Let us do that again, deliberately this time, so the move has a shape you can reuse.

1:42One bar between two children, against four bars among seven children. One half against four sevenths. Different steps again, so nothing can be read off yet. Fourteenths will hold both: halves fit into fourteenths, and so do sevenths. One half becomes seven fourteenths. Four sevenths becomes eight fourteenths. Eight against seven. Four sevenths wins. The move is always the same. Find a step size both fractions can be written in, rewrite both, then compare the counts.

2:20Two facts about sharing sit underneath all of this, and they are worth having in words. The first. If the amount of food is fixed and you add more children, every plate gets smaller. That is not arithmetic, that is what sharing is. The same cake cut for more people gives thinner slices. The second is its mirror image. If the number of children is fixed and you add more food, every plate gets bigger.

2:50Both hold for any numbers at all, and every calculation we do has to agree with them. They are also the fastest way to rank a pair that already shares one of its two numbers. Take the first case. Four bars among seven children, against four bars among eight children. The food is the same. The second group has one extra child. So the second group's plates are smaller, and four sevenths is bigger than four eighths. No conversion, no arithmetic — the sharing tells you.

3:28Now the second case. Same children, more food. One fifth against two fifths: two fifths is bigger. Three sevenths against four sevenths: four sevenths. Be careful with the next one. One half and five eighths look like they should follow the same pattern, but halves and eighths are different steps. That pair still needs converting. Four eighths against five eighths — and five eighths wins. When neither number matches, you go hunting.

4:02Compare three quarters with seven tenths. Write out the other names for three quarters: six eighths, nine twelfths, twelve sixteenths, fifteen twentieths. And for seven tenths: fourteen twentieths, twenty-one thirtieths, twenty-eight fortieths. Both lists contain twentieths. That is the shared step we were hunting for. Fifteen twentieths against fourteen twentieths, so three quarters is the larger. We could have carried on and met again at fortieths — thirty against twenty-eight, the same verdict. Any shared step gives the same answer, so stopping early is economy, not error.

4:47But how do you know the hunt will end at all? Because there is always one shared step waiting, and you can name it without searching. Multiply the two bottom numbers together. Four times ten is forty, and forty is a step size both quarters and tenths fit into — because forty is four lots of ten, and also ten lots of four. That works for every pair, every time. The product of the two bottom numbers is a step both can use.

5:20It is a guarantee, not an instruction. Something smaller often turns up first, as twentieths did. Take the smaller one when you spot it, and take the product when you do not. There is a shorter way of writing the same computation, and once you see where it comes from it stops looking like a trick. Compare four fifths with seven ninths. Over forty-fifths, four fifths becomes thirty-six and seven ninths becomes thirty-five.

5:53Now look at where those two numbers came from. Thirty-six is four times nine. Thirty-five is seven times five. Each top number, multiplied by the other fraction's bottom number. So you can skip writing the shared step at all. Multiply across, compare the two answers, and you have your verdict. It is the same conversion, with the bottom number left unwritten. Pause here and find the shared step for a few pairs yourself.

6:27Seven halves and three fifths. Eight thirds and five sixths. Three quarters and three fifths. One tenth and two ninths. For the first, tenths. For the third, twentieths. For the fourth, ninetieths. The second one deserves a longer look. Three and six — you do not need eighteenths here. Thirds already fit into sixths, so sixths will hold both. Eight thirds becomes sixteen sixths, and five sixths stays exactly as it is. Sixteen against five.

7:02Whenever one bottom number divides the other, the larger of the two is already the shared step. Two proper comparisons now, worked end to end. Four fifths against seven ninths. Five and nine share no factor, so forty-five is the shared step. Thirty-six forty-fifths against thirty-five forty-fifths. Four fifths wins, by one forty-fifth. That is a very narrow margin, and it is exactly why you convert instead of guessing. Now seven ninths against seventeen twenty-firsts. Multiplying the bottoms gives one hundred and eighty-nine, which is large and unnecessary.

7:48Nine and twenty-one both divide sixty-three. Over sixty-three: forty-nine against fifty-one. Seventeen twenty-firsts wins. Narrow again, and again invisible without doing the conversion. So here is the whole method, in two steps. One. Rewrite both fractions so that they use the same step size. Two. Compare the counts. The larger count is the larger fraction. That is all of it. Everything else in this video is either a shortcut for step one or a reason to trust it.

8:28And notice what the method never does. It never compares two top numbers on their own, and it never compares two bottom numbers on their own. Twelve fifths beats eight fifths, because there the steps already match. But nine quarters does not beat five halves, even though nine is much bigger than five. The top number alone decides nothing. Finally, more than two at once. Put four fifths, three quarters, seven tenths and one half in order, smallest first.

9:03You do not need six separate comparisons. Do step one once, for all four together. Twentieths hold every one of them: sixteen twentieths, fifteen twentieths, fourteen twentieths, ten twentieths. Now just read the counts. Ten, fourteen, fifteen, sixteen. So the order is one half, seven tenths, three quarters, four fifths. One conversion, four fractions, and no guessing anywhere. And that is the thread through all of it. Two counts can only be ranked when they are counting the same thing.

9:41Next time: adding and subtracting fractions — which begins with exactly the same first move.