Ratios of three or more quantities at once

0:00Here is a spice rub, ground from four things at once. Eight spoons of paprika, four of pepper, two of garlic, one of salt. Written as a comparison, that is eight to four to two to one. Four terms, in the order the ingredients were named. And the order is part of the information. Swap two of the numbers and you have a different rub. Now, a four-term ratio looks like four things to keep track of. It is really one number waiting to be found.

0:34Start by asking what it actually promises. Paprika to pepper is eight to four, which is two to one. Pepper to garlic is two to one. Garlic to salt is two to one. But that is only the pairs sitting next to each other, and four terms make six pairs, not three. Paprika to garlic is eight to two, which is four to one. Pepper to salt is four to one. And paprika to salt is eight to one.

1:04So a ratio only comparing its neighbours is right about three of the six and drops the other three entirely - first against third, first against fourth, second against fourth. That is half of what the ratio said. If you cannot produce eight to one for the paprika and the salt, you have not read it. Now somebody wants the same rub but only has two spoons of pepper. The recipe calls for four. Two divided by four is one half, and that one number is the whole answer.

1:40Halve everything. Paprika down to four, pepper to two, garlic to one, salt to half a spoon. Four to two to one to nought point five. And check it: all six promises survived. Every pair is exactly what it was. It is the same rub in a smaller quantity. Why does the factor have to reach every term? Watch what happens when it does not. Suppose the salt got left alone. Four to two to one to one.

2:13Three of the six promises are untouched, because they never mentioned the salt - paprika to pepper is still two to one, pepper to garlic still two to one, paprika to garlic still four to one. But the three that do mention it have all halved. Garlic to salt has gone from two to one down to one to one. Pepper to salt from four to one to two to one. Paprika to salt from eight to one to four to one.

2:44That rub is now twice as salty relative to everything else in it. And this is not something about the salt. Every one of the four terms sits in exactly three of the six pairs, so any single term stepping out of line breaks exactly half the ratio - and exactly the half it belongs to. There is a compact way to write what just happened. Eight to four to two to one, double colon, four to two to one to nought point five. Two mixtures, in proportion, term for term.

3:20And what that abbreviates is a chain: eight over four, four over two, two over one, and one over nought point five. Every one of those is two. Not roughly two. Exactly two, four times over. And that shared value is the factor. The chain is usually written out and its common value is left unsaid, which is a shame, because the common value is the entire point. Run the same chain against the salty batch and it gives two, two, two - and then one. It does not settle. That is the test, and that is what failing it looks like.

4:03Now about that nought point five. There is a strong instinct that ratio terms have to be whole numbers, and it is simply wrong. Half a spoon of salt is a real quantity you can measure out. Nothing about a ratio requires its terms to be whole. A decimal in a ratio is not an error to be cleaned up. It is a term, and it behaves like every other term.

4:32That said, a decimal is often telling you something about the order you met things in, rather than about the recipe. Double every term of four to two to one to nought point five and you get eight to four to two to one straight back. So the half was an artefact of which batch happened to be named first. The recipe never had one. Every ratio has a whole-number twin like that. Clear the denominators, take out the common factor, and there it is.

5:05Concrete is a nice case. Cement, sand and gravel at one to one and a half to three - and its twin is two to three to six, the same recipe with nothing to clear. And a batch of twenty, thirty and sixty units is that same recipe again, scaled by ten. The whole-number form turns up on its own if you make enough of it. Now the same method on a different problem, with the known quantity in a different place.

5:37A shade of purple is fixed by red to blue to white in the ratio two to three to five. Somebody has ten litres of white paint and wants to use all of it. Notice where the known quantity sits. In the rub it was the second term. Here it is the last one. Neither position is special, and there is nothing to rearrange. The method does not care which term you were handed.

6:06Here are the two steps, and there are only ever two. First: divide the amount you have by the term that names it. Ten litres of white, against five parts of white, is two litres to the part. That two is the factor, and it is the only thing you had to find. Second: multiply every term by it. Red is two parts, so four litres. Blue is three parts, so six litres. White is five parts, so ten - which is the paint you started with, and a useful check that nothing went wrong.

6:42One division to find the part. Then one multiplication per ingredient. That is the whole procedure, for a ratio of any length. How much purple does that make? Four and six and ten is twenty litres. But look at where that twenty came from. It came out at the end, from the batch. It was never in the ratio. Take the same shade of purple and make four different batches of it - and they total ten litres, twenty, forty and two hundred.

7:17Four different totals, and every one of them keeps every promise the ratio made. All four are the same purple. The recipe's own terms happen to add to ten, which is right for exactly one of those four batches and wrong for the other three. So there is no total until somebody names a quantity. That is exactly why a question like this always hands you one. One more, and it has a term that will bother you.

7:49Concrete for a pillar: cement to sand to gravel at one to one and a half to three. Somebody has three bags of cement. Three bags against one part is a factor of three. So multiply through. Cement three bags, sand four and a half, gravel nine. Four and a half bags is not a mistake and it is not a rounding. The recipe had a half in it before any scaling happened, and the scaling simply carried it through.

8:20And the total is three plus four and a half plus nine - sixteen and a half bags of mixture. So how much does a four-term ratio actually tell you? Not four numbers. Three. Fix how the second, third and fourth compare with the first and the ratio is completely determined, because scaling all four together changes nothing at all. A three-term ratio carries two. A two-term ratio carries one. Always one fewer than it looks like.

8:55And that is why one measured quantity is exactly what finishes the job - no more and no less. The recipe was one number short of a batch, and the measurement is that number. A ratio is not a list of amounts. It is a shape, and it has no size until you give it one.