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:
Multiplying both sides by x:
And dividing both sides by 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.
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!