The distributive property, and using it to compute faster

0:00Two friends sit down to eat, and they order exactly the same thing. A hot dish costing forty three, and something sweet costing twenty four. Each of them orders one of each, so there are two hot dishes and two sweets on that table. What does the bill come to? Here is an expression somebody might write down. Two times forty three, plus twenty four. It has two terms. Two times forty three, and twenty four.

0:30Which is two hot dishes, and one sweet. But there are two sweets on that table, not one. So that line is a correct total for a different order, which is exactly why it is tempting. It comes to a hundred and ten, and it is short by twenty four. One whole sweet, simply missing from the total. There are two honest ways to write what is actually on that table, and both of them work.

1:00The first. One person's order costs forty three plus twenty four. Sixty seven. And there are two people, so two times sixty seven. A hundred and thirty four. The second way is to pay for both hot dishes together first, and then both of the sweets. Two times forty three, plus two times twenty four. Eighty six, plus forty eight. A hundred and thirty four again. Two lots of a total, or two lots of each part added up. The same money.

1:34And none of that depended on there being two of them. With three friends it is three times sixty seven, or three times forty three plus three times twenty four. Two hundred and one, either way. Here is the same idea with something you can actually count. A parade. One group marches in four rows of five. Behind them, a second group marches in three rows of five. How many people are marching in that parade altogether?

2:07Count it as two groups. Four rows of five is twenty. Three rows of five is fifteen. Twenty and fifteen is thirty five. Now count the same parade as one block. Four rows and three rows is seven rows. Seven rows of five. Thirty five. Nobody joined the parade and nobody left it in between the two counts. It is one crowd counted two ways, so the answers were never going to disagree.

2:38That is the entire property, right there. Everything else we do today is that one picture, written down in symbols. Now the boundary of all this, because it does have one, and it is where people go wrong. Five times four, plus three. And five times, bracket, four plus three. Those look almost the same. They are not. Five fours is twenty, and plus three is twenty three. Four plus three is seven, and five sevens is thirty five.

3:13Twenty three against thirty five. Twelve apart. The difference is that in the second one the three got multiplied by five as well, and in the first one it did not. So the factor has to be multiplying the whole bracket. A number simply added on afterwards is not part of it. Those two lines agree only when the multiplier is one, or when the number added on is zero. Any other time they differ.

3:45Let us see why this keeps working, rather than just that it does. Take ninety eight, and write it out ten times in a row. Now write three more ninety eights on the end of that same row. The first run is ten ninety eights. Nine hundred and eighty. The second run is three ninety eights. Two hundred and ninety four. Put the two together and you have twelve hundred and seventy four.

4:11But now cover up the braces, and look at the whole line as one single row. It is just ninety eights, one after another. Count them. Thirteen. Thirteen ninety eights, and twelve hundred and seventy four. Ninety eight times ten, plus ninety eight times three, is ninety eight times thirteen. Nothing moved. We only drew the brace somewhere else. Now the same argument, but with a subtraction in it. Write out fourteen tens.

4:45That comes to a hundred and forty. Now strike out the last six of them. Six tens gone. Sixty. So what is actually left on the line now? A hundred and forty take away sixty. Eighty. And counting what survived instead: eight tens. Eighty. Ten times fourteen, minus ten times six, is ten times eight. It holds for a difference exactly as it did for a total, and for the same reason. We struck things out. We did not change what they were.

5:22So here is the whole thing, stated in one line. Multiplying a total gives the same answer as multiplying each part and then adding. And multiplying a difference gives the same answer as multiplying each part and then subtracting. That is called the distributive property. It is worth being careful about what that sentence is claiming. It is not a rule about brackets. It is a statement about counting one collection two ways.

5:55The parade was not obeying a law when it came out the same twice. It was just sitting there being thirty five people. Which means you never have to remember this. You can rebuild it any time from seven rows of five. And now we can turn it round and use it. Because if two expressions are equal, you are allowed to pick whichever one is easier to work out.

6:22Suppose somebody tells you that fifty three times eighteen is nine hundred and fifty four. And then, without any warning at all, asks you for sixty three times eighteen. You could start again from the beginning. Or you could notice that sixty three is fifty three plus ten. So sixty three eighteens is fifty three eighteens, and then another ten eighteens. You already have the fifty three eighteens. Nine hundred and fifty four.

6:54And ten eighteens is a hundred and eighty. Nine hundred and fifty four, plus a hundred and eighty. Eleven hundred and thirty four. That is one addition, instead of a second two-digit multiplication. The product you already knew did most of the work, and the property is what let you use it. Now the move that is genuinely worth having. Ninety seven times twenty five. That is genuinely unpleasant to do the long way, and it does not have to be.

7:29Ninety seven is a hundred, less three. So ninety seven twenty fives is a hundred twenty fives, less three twenty fives. A hundred twenty fives is twenty five hundred. Three twenty fives is seventy five. Twenty five hundred, take away seventy five. Twenty four hundred and twenty five. One awkward multiplication became one you already knew and one you can do in your head. So should you always do this? No. And here is how to tell.

8:04Try it on forty seven times eight, and split the forty seven as twenty three plus twenty four. That gives you twenty three eights, plus twenty four eights. That is two hard multiplications where you started with one. The split made it worse. Now split the same forty seven as fifty minus three. Fifty eights is four hundred. Three eights is twenty four. Four hundred take away twenty four is three hundred and seventy six. Correct, and much easier.

8:38So it was never the number that was the problem. It was the split. And the test is simple. Is each piece something you can multiply by in one step? A single digit, or a round number. Every two-digit number has a split like that waiting in it. The only skill is seeing which one.