Tutoring For All Ages

Here’s today’s medium-level SAT Question of the Day (note that if you saw it on the SSAT/ISEE, it would be a Hard question):

In the figure above, the large rectangle is divided into six identical small squares. If the perimeter of the large rectangle is 30, what is the perimeter of one of the small squares?

A) 5
B) 8
C) 9
D) 10
E) 12

How do you solve this problem in the quickest way possible?

Here’s how the College Board thinks you should solve it. That’s the “right” way to do it: use the ratio of length to width to set up an equation using the overall perimeter. And assuming you don’t make a careless algebra error, that will get you to the right answer. For some students, it may even be the fastest way to get to the right answer.

But not every student is going to see this way immediately. If you’re one of them, consider another way: working backwards, sometimes called guess-and-check. Start with the middle answer choice and assume it’s the perimeter of a square. Figure out what each side of the rectangle then has to be, and see if that matches the given perimeter of the whole rectangle. If it does, you have your answer. If not, you should be able to tell whether you need the perimeter to be bigger or smaller, so you can figure out which answer choice to try next.

In this case, you might also want to consider the fact that you’re starting with a perimeter, and the first thing you’re going to have to do in this method is divide by 4. All the answers are integers, and the given perimeter is also an integer, and the test-makers like to use integers on medium-level SAT questions (which this is). So, while it’s not a guarantee that the right answer (the perimeter of a 4-sided figure) will be divisible by 4, it’s a pretty good bet. And if it’s true, it means you can eliminate three answer choices. So instead of starting with answer choice C, try starting with B. If the perimeter of the square is 8, each side is 2, which means the rectangle is 4×6 and its perimeter is 20. That’s way too small, so try the same thing with the other answer that’s divisible by 4, and the answer works out correctly.

If you don’t like working backwards but you also don’t like the SAT’s method, try a bit of picking numbers: what if the side of each square were 1 (the simplest guess)? That’s a perimeter of 10, which is 3 times too small, so each square must be a 3×3, leading to a square perimeter of 12.

There’s almost always more than one way to solve a standardized test problem.