Brahmagupta's method: you can only add units of the same size

0:00A bar of chocolate, and two people eating from it. Meena eats one half of the bar. Her brother eats one quarter. How much of the bar did the two of them eat between them? You can see the answer coming, but watch what your eye actually does to get there. It does not add one and one, and it does not add two and four. It waits until the pieces are the same size, and then it counts them.

0:28That is the whole of this video, and everything else is detail. So let us do exactly that, slowly. The half and the quarter are different sized pieces, so there is nothing to count yet. Cut the half down the middle. Now it is two quarters — the same amount of chocolate, cut smaller. And now every piece on the bar is a quarter. Two quarters eaten by Meena, one quarter by her brother. Two and one is three.

0:59Three quarters of the bar, between them. Notice that the quarters never changed size while we counted. Only the count changed. That condition is the one thing the whole method exists to protect. Before we go any further, there is a second question sitting right there. How much of the bar is left? One piece out of the four. One quarter. And look at what you just did to get it. You took three quarters away from the whole bar.

1:31That is a subtraction, and you did it without being taught one. It is the same move in reverse — same sized pieces, then count. Which is why adding and subtracting fractions are one topic and not two. Here is the plainest possible case, so the idea has nothing to hide behind. Two fifths, drawn as two shaded blocks of five. And one fifth, one shaded block of five. Slide them together. Three shaded blocks. Three fifths.

2:06The bottom number did not move, and it should not have. It says what size the blocks are, and nobody resized anything. Two fifths plus one fifth is three fifths, for the same reason two apples plus one apple is three apples. The bottom number is the name of the thing you are counting. That is not a rule to memorise. It is what the picture already does. Now a sum where something slightly alarming happens.

2:39Four sevenths plus six sevenths. Same sized blocks, so we can go straight to counting. Four and six is ten. Ten sevenths. But a whole is only seven sevenths, so ten sevenths is more than one whole bar. That is not a mistake. It just means the two amounts together filled a bar and spilled into a second one. Ten sevenths, or one whole and three sevenths. The same number, written two ways.

3:10An answer bigger than one whole is not a warning sign. It is just a number. So when is adding fractions actually hard? Only when the blocks are different sizes. One quarter and one third. Put the two strips side by side and look at them. There is no honest way to count these. A quarter and a third are not the same object. And here is the mistake almost everybody makes at this point: adding the tops and adding the bottoms.

3:43One plus one over four plus three. Two sevenths. But two sevenths is smaller than one third — it is smaller than one of the two things we started with. An amount cannot shrink by having something added to it. Whatever that operation is, it is not addition. The fix is the one you already know from comparing. Find a step size both fractions can be written in. Quarters and thirds both fit into twelfths.

4:15One quarter becomes three twelfths. One third becomes four twelfths. Now the blocks match, so now we may count. Three and four is seven. Seven twelfths. Written as a procedure, that is three steps. Find a shared unit using a common multiple of the two bottom numbers. Add the counts, and keep that bottom number. Then shorten the answer if you want to. This method was written down in general form in the seventh century, by the mathematician Brahmagupta.

4:52Two more sums, because choosing the shared unit is where the judgement lives. Two thirds plus one fifth. Three and five share no factor, so fifteenths it is — which is both the product and the smallest choice. Ten fifteenths plus three fifteenths is thirteen fifteenths. Now one sixth plus one third. Multiply the bottoms and you get eighteen, and eighteenths would work. But look again. Three already divides six.

5:25So sixths will hold both, and one sixth does not have to change at all. One sixth plus two sixths is three sixths. Smaller numbers, less writing, same answer. Three sixths, though, has a shorter name. Three and six are both divisible by three, so three sixths is one half. Watch the number line while that happens. The point does not move. Shortening the answer is tidying it, not finishing it. The sum was already done.

5:58Had we gone the long way round and used eighteenths, we would have landed on nine eighteenths — which is also one half. Different route, different writing, same point on the line. In this series we will shorten answers when they shorten easily, and leave them alone when they do not. Subtraction now, and there is genuinely less to say, because it is the same method. Six sevenths, drawn as six shaded blocks. Take four of them away.

6:31Two blocks left. Two sevenths. The bottom number stayed at seven throughout, for the same reason as before — nothing was resized. Three quick ones, and notice that tidying is a separate question from subtracting. Five eighths less three eighths is two eighths, which tidies to one quarter. Seven ninths less five ninths is two ninths, which does not tidy — two and nine share nothing. And ten twenty-sevenths less one twenty-seventh is nine twenty-sevenths, which is one third.

7:08And when the sizes differ, you already know what happens first. Three quarters less two thirds. Twelfths hold both. Three quarters is nine twelfths, because four times three is twelve. Two thirds is eight twelfths, because three times four is twelve. Nine take away eight is one. One twelfth. A very small answer, and it should be — three quarters and two thirds are nearly the same size. The steps are the addition steps with one sign changed, which is exactly why it was never introduced as a new method.

7:47Last, the version that turns up in real life, where the arithmetic is buried in a sentence. A painter mixes two thirds of a litre of one colour with three quarters of a litre of another. Over twelfths that is eight plus nine, so seventeen twelfths — one and five twelfths of a litre. Two people buy ribbon for a border one metre long: two fifths of a metre and three quarters of a metre.

8:17Over twentieths, eight plus fifteen is twenty-three twentieths. That is more than a metre, so yes, there is enough — with three twentieths to spare. Answer the question that was asked, not just the sum inside it. One last trap. Two runners take ten thirds of a minute and thirteen quarters of a minute for a lap. Who is faster? Over twelfths that is forty against thirty-nine, so thirteen quarters is the smaller time — and the smaller time is the faster runner, by one twelfth of a minute.

8:54Next time: where these fraction symbols came from, and the people who first wrote them down.