Direct proportion restated: the quotient that stays constant

0:00Two cooks are making pancake batter, and both are working from the same mixing rule. The first uses six cups of flour and three cups of milk. The second uses four cups of flour and two cups of milk. Everything else is the same. Will they taste alike? You cannot answer that from the amounts, because the amounts are different. Six is not four, and three is not two. Whatever settles this is not about how much. It is about how much compared to how much.

0:36So write each batch as a comparison instead. Six to three. Four to two. And the mixing rule they are both working from is two to one. Two parts flour for every one part milk. Notice what that is. Two to one is not an amount of anything. You cannot pour it. It is a rule for pouring, and it holds at any size. So the word we want is not equal. These two batches are not equal and never will be.

1:07They are proportional, which is a different word doing a different job. Same recipe, different quantity. Here is the first way to read it, and it goes down each batch on its own. Six cups of flour divided by three cups of milk is two. Four divided by two is also two. Both cooks are at two parts flour per part milk. That is the mixing rule, recovered from the amounts.

1:38And notice the units. Cups on the top, cups on the bottom, and they cancel. So the two is a plain number with nothing attached, which is why it can be compared between two kitchens. Now here is the second way, and it goes across, comparing one batch against the other. Six cups of flour against four cups of flour. Six divided by four is one and a half. Three cups of milk against two cups of milk. Three divided by two is also one and a half.

2:14One and a half, twice. The first batch is one and a half times the second, in both ingredients at once. That is what makes it a scaled copy rather than a different recipe. One number carries the whole thing across. And it is exact. Three halves, not a decimal rounded on the way past. Stop and look at what just happened, because both of those are true at the same time.

2:43One reading says two. The other says one and a half. They are not the same number, and not measuring the same thing. Two is how the recipe is built. One and a half is how big the batch is. They even divide different things, and you can catch them out on it. Ask the first reading about a batch with no milk and it falls over, because you cannot divide by nothing.

3:09Ask the second reading and it answers quite happily. Swap which batch is missing the ingredient, and the two of them swap places. So these are two different sentences. And yet they always give the same verdict. Why? Because underneath, they are the same sentence. Take the first reading: six over three equals four over two. Multiply both sides by three, then by two. The threes cancel on the left, the twos on the right, and what is left is six times two equals three times four.

3:48Now the second reading: six over four equals three over two. Multiply by four, then by two. The fours cancel and the twos cancel, and what is left is six times two equals four times three. The same line. Twelve on one side. Twelve on the other. Both readings, cleared of their division, land on one product equality. That is cross-multiplication. Not a trick somebody invented. It is what is left when you take the dividing out.

4:25And that is the form to reach for, for two reasons. The first is that there is no division in it at all, so there is nothing that can go wrong with dividing. The second is that you do not have to decide which reading you meant. It is what both of them agree on, so it answers for both. But there is one case where it stops being a test, and it should be said out loud.

4:51Put nothing against nothing, beside three to five. Nothing times five is nought. Nothing times three is nought. The products match, so the test says yes. And nothing to nothing is not a recipe at all. There is no rule in it to compare. So: nothing to divide by is a convenience, and it is not the same as settling a missing quantity. Keep those apart. There is a picture for the product line, and it is worth drawing carefully.

5:25Write the two comparisons side by side. Six, three, four, two. Arch a curve over the top from the six across to the two. Those two multiplied give twelve. Now dip a second curve underneath from the three across to the four. Those two multiplied also give twelve. The two curves are nested, one inside the other, and they never cross. Each one is picking out a first term of one comparison with a second term of the other. That is the pairing, and that is all cross-multiplication ever was.

6:04Two ways of thinking a comparison is being kept, both of which are not. The first: add the same amount to both. Six and three, add one cup to each, and you get seven to four. Test it. Six times four is twenty-four. Three times seven is twenty-one. They disagree, so the recipe changed. Multiply instead. Double both and you get twelve to six. Six times six is thirty-six, and three times twelve is thirty-six. They agree.

6:38Comparisons are kept by multiplying, never by adding. The second: same difference means same comparison. Six to three has a gap of three, and so does eight to five. But six to three is two parts to one, and eight to five is not. Put that idea against six real pairs and it gets the answer right once out of six, and the one it gets right is a no.

7:08Now watch the test earn its keep on a batch that does not pass. Suppose the second cook had used four cups of flour and three of milk instead. Six times three is eighteen. Three times four is twelve. Eighteen against twelve, so no. And the test tells you which way it went wrong. Four over three is less than two, so there is too much milk for the flour.

7:34There are exactly two ways to repair it, and the same line gives you both. Hold the flour at four and the milk has to come down to two. Or hold the milk at three and the flour has to go up to six. One test, two repairs, and no guessing in between. The same line does one more job, and this is the one you will actually use. Five workers shift four and a half thousand bricks in a day. You need eighteen thousand shifted in a day. How many workers?

8:09Set it up as four and a half thousand is to eighteen thousand, as five is to the number you want. Look at what is compared against what. Bricks against bricks. Workers against workers. That matters, because it means both comparisons are plain numbers with no unit left on them, and only then can they be set equal. Cross-multiply, and the answer is twenty workers. Read it either way and it checks out. Across: eighteen thousand is four times four and a half thousand, and twenty is four times five. Down: nine hundred bricks per worker at both ends.

8:53One last thing, and it is the honest part. The test told you that flour and milk are in the same comparison in both batches. That is all it told you. Suppose the first cook also put in two spoons of sugar and five of butter, and the second put in one of each. Flour and milk still test exactly as before. Nothing there changed. But two to five is not one to one, so the sugar and butter are nowhere near matching, and those pancakes will not taste the same.

9:28Two ingredients matching is a claim about two ingredients. Everything you did not name is free to be anything at all. A strong tool, and a narrow one. Say what you compared, and you will know what you have proved.