Comparing two expressions by reasoning, not by evaluating

0:00You already know what these three signs do between two numbers. Less than, greater than, equal to. Here is something slightly less familiar. You can put them between two expressions. Ten plus two, greater than, seven plus one. There is no plain number written anywhere on that line, and it is still a perfectly ordinary statement to make. Because the sign is not comparing how the two sides look. It is comparing what they stand for. Their values.

0:32Twelve is greater than eight, so that sign is correct. But notice we had to work both sides out to check it. For the rest of this video, we are going to try very hard not to do that. First, one that makes the point properly. Thirteen minus two, against four times three. Those two do not look remotely alike. One is a subtraction, the other a multiplication. One of them has bigger numbers in it. One of them is longer to write.

1:06None of that decides anything at all. Thirteen minus two is eleven. Four times three is twelve. So the subtraction is the smaller one, and the sign points that way. The bigger-looking side lost, and that will keep happening. The sign records an ordering of values, and values are the only thing it sees. So: can we ever get that ordering without finding the values first? Here is a pair where we can.

1:37One thousand and twenty three, plus one hundred and twenty five. Against one thousand and twenty two, plus one hundred and twenty eight. Please do not add those up. Just look at them. Now let me tell you the same thing as a story. Ana has a jar of marbles, and somebody gives her some more. Ben has a jar of marbles, and somebody gives him some more. Ana started with one thousand and twenty three, and was given one hundred and twenty five.

2:11Ben started with one thousand and twenty two, and was given one hundred and twenty eight. Who ends up with more marbles? The trick is to line the two of them up part against part, and then look only at what actually differs. Start with what they started with. Ana had one thousand and twenty three. Ben had one thousand and twenty two. So they had the same amount, except Ana had one extra marble.

2:42Now what they were given. Ana got one hundred and twenty five. Ben got one hundred and twenty eight. Same amount again, except Ben got three extra. That is the whole comparison. Ana is one ahead. Ben gains three. Everything else on those two lines is identical, and identical things cancel. So we can throw all of it away and keep two small numbers. So: Ana starts one ahead. Ben gains three more than she does.

3:16Three is bigger than one, so Ben must finish in front. By how much? Three minus one. Two marbles. And there it is. Ben ends up two ahead of Ana. So the first expression is the smaller one, and we can write that sign down with confidence. Now notice what we never did. We never added one thousand and twenty three to one hundred and twenty five. We never added one thousand and twenty two to one hundred and twenty eight.

3:48Neither total was needed. The comparison was decided by the difference alone. Now the same idea, but with things being taken away, and this is where it gets slippery. One hundred and thirteen, minus twenty five. Against one hundred and twelve, minus twenty four. As a story: Ana had one hundred and thirteen marbles and lost twenty five of them. Ben had one hundred and twelve, and lost twenty four.

4:20Line them up the same way. Ana started with one more. And Ana lost one more. So she is one ahead at the start, and one worse off in what she lost. Your instinct might be that starting ahead and losing more roughly balances out. It does not roughly balance out. It balances out exactly, and the two of them end level. That is worth slowing down on, because a lot of mistakes live here.

4:51Adding one to what you start with pushes the answer up by one. That much is obvious, and nobody has ever got it wrong. Adding one to what you take away pushes the answer down by one. That is the one people get wrong, and it is worth saying slowly. A bigger amount removed means a smaller amount left over at the end. So the very same change, plus one, moves the answer in opposite directions depending on where it lands.

5:24Ana's extra marble at the start pushed her up by one. Her extra loss pushed her down by one. Up one, down one, and they cancel to nothing. Which is why those two expressions are exactly equal, and neither total was ever worked out. So let us run the method. Five pairs, and for each one, only the shifts. Two hundred and forty five plus two eighty nine, against two forty six plus two eighty five.

5:55First part up one, second part down four. Net effect, down three. So the left side is bigger. Two seventy three minus one forty five, against two seventy two minus one forty four. Started with one less, took away one less. Down one, then up one. They are equal. Three sixty four plus five eighty seven, against three sixty three plus five eighty nine. Down one, up two. Net up one, so the right side wins by one.

6:28One twenty four plus two forty five, against one twenty nine plus two forty five. Second parts identical, first part up five. Right side, by five, and there was nothing else to think about. And the last one, which is the interesting one. Two hundred and thirteen minus seventy seven, against two fourteen minus seventy six. The starting amount goes up by one. The amount removed goes down by one.

7:01Now compare that with the second pair we just did. In that one, the amount removed also went down by one. Exactly the same change. But there, the starting amount went down as well, and the two cancelled. Here, the starting amount goes up instead. So the two changes agree with each other. Up one from starting higher, and up another one from removing less. They add up rather than cancel, so the right side wins by two.

7:32Same shift in the removed part, opposite verdicts, and the only difference is what the other part did. Which is why you have to track both, and why the direction of each one matters. One last thing, and it is the honest part. This does not always work, and it is worth knowing when to stop. Here are five expressions, and the job is to put them in order of size.

8:01Sixty seven minus nineteen. Sixty seven minus twenty. Thirty five plus twenty five. Five times eleven. And one hundred and twenty divided by three. Look at the first two. Same starting amount, and the second one takes away more. So the second is smaller. Settled, without touching either. Now look at the other three. There is nothing to line them up against. No shared parts, no matching structure, nothing to cancel. So work them out. Forty. Fifty five. Sixty.

8:38That is not a failure of the method. Reasoning is quicker when there is something to reason about. The question was always whether a comparison can be settled without heavy calculation. Sometimes it can, and knowing which is the actual skill.