In this entry, I introduced a problem solving question and we began to approach the question and build our scratch paper notes.

Now, we need to evaluate the expressions. Unless we are number property rock stars, the best way to do this is to pick numbers to test them. The easiest fraction to pick is 1/2, so let’s go with x=1/2.

You may be tempted to rip through this part at a good clip, but I would encourage you to be careful. You may need to check your work. You may get confused. You will want to confirm your answer at the end, and that will mean doing the problem all over again if you haven’t left yourself those moonstones. That’s especially the case with an all-that-apply question like this, where you won’t find the one right answer and be done, but instead must evaluate each choice on its own merits. Once you’ve evaluated them, write whether they came out true or false, like this:

The calculations here are straightforward, but if they were more complex and you used the calculator, I’d still encourage you to write it out. There’s even more of a chance you’ll lose track of what you are doing when you go back and forth to the calculator, and if you don’t write it down you’ll have no idea what you did. That’s exactly how the calculator can be a trap on test day.

A,B, and C are pretty easy. Doubling 1/2 gives you 1, which disproves all 3. Once you’ve come up with a False answer, you can eliminate it. It doesn’t matter if, as in C, 2x could sometimes be larger than 1 (like if we had picked x=3/4). An expression that is SOMETIMES FALSE is NOT an expression that MUST BE TRUE. So anything that came back FALSE is eliminated once and for all.

We do have to take a second pass at D and E though, to see if these choices are SOMETIMES TRUE or if they MUST be TRUE. Here, we can pick different permissible numbers and test them. We’re pretty sure a number smaller than 1/2 would still be smaller than 1, and itself, but what about a larger one, like 3/4? 3/4 squared is 9/16. That’s less than 1. It’s also less than 12/16, which is 3/4. So D and E stand: we can be sure that they MUST be TRUE, and are the correct answers.

That last step might have triggered us to remember the number property that when we square a positive fraction between 0 and 1, that number gets SMALLER. Ultimately, that is the number property that is being tested here. A savvy test taker is thinking at that point, “I knew it, you evil stepmothers, you can’t fool me!”.

You can save some time if you can recognize these number properties; but if you can’t remember them or don’t see it, that’s ok. Picking numbers works just as well, because numbers have to behave like numbers. Either way, careful scratch work saves the day on thorny logic problems like this one.