Marlo travels 17 yards every x seconds. How many seconds will it take him to travel y yards at the same speed?

A student I was working with encountered a very similar problem to this one and proceeded to make a mess of it. I took a look at his algebra, and slowly tried to figure out where the error had entered the equation. It took a full minute of careful searching to find the error, and even when I explained what had gone wrong all I got was a blank expression in return.

Completing this problem algebraically starts with two equal rates (where z is the variable we’re trying to solve for in number of seconds):

17/x = y/z

Multiplying both sides by z:

17z/x=y

Multiplying both sides by x:

17z =xy

And dividing both sides by 17:

Z=xy/17

The real challenge here is in getting the initial equation set up correctly, because the manipulations from there are pretty straightforward. But, rather than give an in-depth algebra lesson to my student, my question was why? Why even mess with the algebra? Why go down a path so fraught with peril if you know that algebra isn’t a huge strength of yours? The part of this problem that I haven’t given you, and the part that makes all the difference is the answer choices.

A) 17xy

B) 17x/y

C) 17/xy

D) x/17y

E) xy/17

Since we’re given variables in the answer choices, rather than deal with them and their potential for confusion, let’s instead plug our own values in for x and y, figure out the answer and then see what matches.

For instance, let’s say Marlo travels 17 yards every 5 seconds. We’ve now defined that x=5. If y=34 yards, we can easily calculate that it will take Marlo 10 seconds to travel that distance at the same speed. So, when we pick x=5 and y=34, the answer should be 10. Let’s test the answer choices to find what’s consistent with our choice.

A) 17xy ; 17(5)(34) = Way more than 10, not correct

B) 17x/y ; 17(5)/(34) = 2.5, too small. Eliminate this one.

C) 17/xy ; 17/(5)(34) = Much smaller than 10, move on.

D) x/17y ; (5)/(17)(34) = Much too small again

E) xy/17 ; (5)(34)/17 = 10. There we go!

There are no hero points for taking on complicated algebra equations on a multiple choice test. Let this example show you that when you have variables in your answer choices picking numbers is a great strategy to get you to the correct answer… whether your math teacher would have done it that way or not!