ACT Math: Looking for the “Clever” Solution

The Math portion of the ACT tests how well you’ve mastered the math you’ve been taught in middle school and high school. But on some questions, it does more than that. There are some questions that test how deeply you know the material, how good you are at finding an efficient solution to a difficult problem. In a sense, these problems test how “clever” you are.

Consider the following problem:

If xy = 28 and x + y = 7, what is the sum of the reciprocals of x and y?

A.  1/28
B.  1/7
C.  1/4
D.  4/7
E.  3/4

The “Math Class” Solution
Let’s start by doing this problem the way we would if we were in math class at school. It’s really just a systems of equations problem. We can solve the second equation for x or y, do a little substitution and voila! … out come the values of x and y. Doesn’t seem that bad, right?

If xy = 7, then it follows that y = 7 – x. Let’s substitute that into the first equation and solve for x.

xy = 28
x(7 – x) = 28
7x – x² = 28
0 = x² – 7x + 28

We’d try to factor that …. but, darn! It doesn’t factor. That means we’ll need to use Quadratic Formula. And right about now, you’re starting to realize that the values of x and y are really ugly and their reciprocals are probably worse.

The “Clever” Solution
So maybe there’s another way to do this, a way that doesn’t even require us to figure out what x and y are. Let’s instead play around with the thing that we’re trying to find

$\frac{1}{x}+\frac{1}{y}$.

To add those, we’ll need a common denominator. In this case it’s xy.

$\frac{y}{xy}+\frac{x}{xy}=\frac{x+y}{xy}$

Aha! Those two expressions in the numerator and the denominator are the two things we were given in the problem! We’re almost done.

$\frac{x+y}{xy}=\frac{7}{28}=\frac{1}{4}$

The correct answer is Choice C.

A lot of you are probably thinking “great … but I’m not sure I’d ever think of that.” And, yeah, I agree that coming up with that approach is pretty tricky. I do a lot of math, so that approach popped into my head. That’s how you’ll get good at recognizing when to do something “clever” as well. You need to practice a lot of problems, and you need to be flexible in your approach. If the traditional approach looks like it’s going to take you a whole bunch of steps or lead you to really ugly answers, consider another approach. Sometimes being clever can get you to the right answer a whole lot faster.

If you have questions about this problem or anything to do with the ACT, send us an email at info@cardinalec.com