Dividing a quantity in a given ratio

0:00Two questions that sound like opposites. Make more of this, keeping the recipe. And: cut this up, keeping the recipe. One builds a quantity outward from a part you were handed. The other carves a quantity you already have into shares. They feel like different jobs. They are the same operation, run in opposite directions. And the hinge in both is one number: the size of a single part. Find that, and everything else is multiplication.

0:34Start small. Divide twelve in the ratio two to one. Two parts of one thing, one part of the other. So how many parts are there altogether? Three. Now hold on to that three, because it is the number people lose. Three is not a quantity. There are not three of anything here. Three is a COUNT of equal parts, and the twelve is what those parts have to share out.

1:01The sum of the terms counts parts. The whole is measured in units. Keep those two numbers apart and this topic is easy. So: twelve units, three parts. One part is twelve divided by three, which is four. That is it. That is the whole calculation. Everything after it is multiplying. Two parts is eight. One part is four. And check: eight plus four is twelve. Nothing went missing and nothing got invented.

1:36Notice the shape of it. Divide once by the sum. Then multiply once per share. Two steps, whatever the ratio is. Now a harder one, and it is harder in an interesting way. A hundred and ten units of concrete are wanted, with cement, sand and gravel in the ratio one to one and a half to three. Add the terms. One plus one and a half plus three is five and a half.

2:06There is a real instinct at this point that you cannot divide by five and a half. You can, and you must. A hundred and ten divided by five and a half is twenty. One part is twenty units. So cement is one part, twenty. Sand is one and a half parts, thirty. Gravel is three parts, sixty. And twenty plus thirty plus sixty is a hundred and ten. Why did that close up so neatly? It was not luck, and it is worth seeing why not.

2:40Look at what each share really is: the whole, times that term, over the sum of the terms. For the concrete those fractions are two elevenths, three elevenths and six elevenths. And two plus three plus six is eleven. The fractions add to eleven elevenths, which is one. That is the whole argument. The fractions are built out of a sum that IS the sum of the terms, so they cannot add to anything but one.

3:11So the shares cannot add to anything but the whole - whatever the whole is, and whatever the ratio is. The rule is not just usable. It is trustworthy. Which gives you the same step written a second way, and you will meet both. Instead of dividing and then multiplying, multiply the whole by this term over all the terms. Cement is a hundred and ten times one over five and a half. Sand is a hundred and ten times one and a half over five and a half.

3:46Same twenty. Same thirty. It has to be, because it is the same two operations in the other order. Use whichever you like. But the dividing form has one advantage: it puts the size of one part on the table where you can see it. Here is the wrong move, and it is worth doing deliberately once. Divide twelve in the ratio two to one. Somebody divides twelve by two, and twelve by one.

4:18Six and twelve. Which add to eighteen. Eighteen. Out of a whole of twelve. The method invented six units of something out of nothing. And that failure is not an accident of these numbers. Put that idea to nine different splittings and it closes on three of them and fails on the other six. The three it survives are all ratios whose terms are every one the same - which is no help on any real mixture. The division is by the SUM, and the check is that the shares add back.

4:56Now a question that looks the same and is not. A library holds picture books, novels and reference books in the ratio three to two to one, and there are two hundred and eighty-eight picture books. Stop before you divide. Two hundred and eighty-eight is not the whole. It is one share. So do not divide by the sum. Divide by the term that names it: two hundred and eighty-eight over three is ninety-six books to the part.

5:28Then multiply through: two hundred and eighty-eight, one hundred and ninety-two, ninety-six - five hundred and seventy-six books in all. Get that wrong and read the share as the whole, and you get a hundred and forty-four, ninety-six and forty-eight. Those add to two hundred and eighty-eight, so the check passes - and every single number is wrong. Put the two directions side by side on one example. A purple is fixed by red to blue to white in the ratio two to three to five.

6:05First question: you have fifty millilitres of finished purple. The terms add to ten, one part is five millilitres, and the shares are ten, fifteen and twenty-five. Second question: you have ten litres of white and want to use it all. White is five parts, so one part is two litres, and you need four of red and six of blue - twenty litres of paint. Same ratio. Opposite directions. And in both, the first thing computed was the size of one part.

6:42Every division so far had a whole that somebody chose. A hundred and ten units. Fifty millilitres. Here is one where the whole is not up for discussion. A triangle's angles are in the ratio one to three to five. What are they? You were never told a total. But you have one anyway, because a triangle's angles add to a hundred and eighty, and that was settled long before the question was asked.

7:12So divide it. One plus three plus five is nine. A hundred and eighty over nine is twenty degrees to the part. Twenty, sixty and a hundred. And they add to a hundred and eighty, because of course they do. Now change one word and watch the whole thing collapse. A triangle whose SIDES are in the ratio one to three to five. What are they? There is no answer, and there are two separate reasons why.

7:43The first: no such triangle exists. One and three together are four, and four cannot reach across five. Two sides have to beat the third. The second is deeper. Take a ratio that does work - three to four to five. Those are the sides of a triangle at any size at all: three, four, five; or six, eight, ten; or thirty, forty, fifty. One shape, three different perimeters. Nothing in the ratio picks one, because there was never a fixed total to divide.

8:19That is the difference. The angle sum was handed to you. A perimeter never is. One last trap, and it catches careful people. A hundred coins, worth ten, five, two and one, in the ratio four to three to two to one. Ten parts, ten coins to the part. Forty, thirty, twenty and ten coins. That much is easy. Now: how much money? Four hundred, a hundred and fifty, forty and ten. Six hundred altogether.

8:54But look at those four amounts as a ratio. Simplified, they are forty to fifteen to four to one - and that is not four to three to two to one. It is not even close. Dividing a quantity in a ratio divides exactly the thing you divided. The counts were in that ratio. Nothing else was promised, and nothing else came true. Find the size of one part. Multiply. Check it adds back. And be precise about what, exactly, you were splitting.