Try Out the Answers!
Attempt the question; then read the explanation below.
Textbook Approach or Solution Strategies?
There are at least two possible approaches to every GRE Quant problem:
- It is always possible to attempt a direct “textbook” solution. This is the more conventional approach, the way you might wish to solve a problem in a high school math class.
- Likewise, it is always possible to use an alternate, “Solution Strategies” approach. These approaches involve less calculation and more observation. Picture yourself as a detective trying to spot the shortcuts or the faster path to the correct answer.
On your actual GRE, you will likely employ a combination of the strategies above. Some problems lend themselves to short, direct solutions. Others are so convoluted or time-consuming that you would be well-served to consider alternate approaches.
This problem of the week is an example of an algebra problem that benefits from using a solution strategy. Before we consider the alternate approach, let’s step through a conventional solution.
We can begin by Recording What We Know.
Our variables are T for the total song count, H for hip-hop, D for dubstep, and P for pop. We have indicated that the total is the sum of the other three variables; there are 18 hip-hop songs; dubstep is one-third of the total; to get P we add hip-hop to dubstep and divide that sum by 2.
We’re trying to solve for T, so let’s use substitution to eliminate multiple variables and create an equation in which T is the only variable.
Start by substituting 18 for H. Next we can substitute T ÷ 3 for D in both the first equation and the equation for P. Next we can substitute (18 + T ÷ 3) ÷ 2 for P in the first equation. Now we have an equation for the total that has only one variable. Great!
Solving for T, we can start by cross-multiplying. Next we can combine our T fractions and our constants. We subtract T from both sides to get our solution, T = 54.
Is There a Faster Way?
While the above approach will get you to the right answer, you might notice that it’s both time-consuming and leaves open the possibility of making mistakes. Sometimes when you’re doing a lengthy algebra problem, you can mix up a variable or lose track of your work. A small mistake can create big problems.
You must avoid mistakes such as these. On a problem like this, one way to avoid mistakes is to evaluate the answers to see what’s going on. Notice that the answer choices here are all possible different totals. What if we could try out these answers to see which one works? How do we do that?
At PowerScore we use an approach called Backplugging. Start by labeling your answer choices. What do they represent?
The answers all represent the total. Let’s start in the middle to narrow things down. Imagine that the total is 48. What’s the next thing we know? Now we already know H is 18. Let’s jot that down in a column. Now let’s solve for D using the total of 48. If D is the total divided by 3, we get D = 16.
Time for another column. Let’s solve for P. P equals the sum of D and H divided by 2. We know D = 16 from the previous step. 16 plus 18 equals 34. 34 divided by 2 is 17.
Check it out! We now have three possible values for H, D, and P. Now we need to see whether these values work. Let’s add them up and see whether they equal 48. No they don’t! The total is too big; they add up to 51. Since H is constant at 18, we should consider making our total bigger. Let’s eliminate answer choice C and go down to answer choice D. See what happens:
We work through the same process one more time. This time we get 18 for D and 18 for P, using the same calculations. This time D + H + P does add up to 54. That’s it. We’re done. This is the correct answer.
Even if D hadn’t been correct, as long as we had been getting closer, we could have inferred that E must be correct.
This Backplugging approach enables you to solve a problem like this in two or possibly three steps. Your work is both clearer and more accurate, less prone to mistakes. Try it out yourself!
Work Every Problem Multiple Ways
A key to success on GRE Quant is never to get stuck. Keep plugging away at the problems. If one approach doesn’t work, try something else or skip the problem and come back to it later. You must maintain your momentum at all times. By practicing multiple approaches to problems, you are equipping yourself with a toolbox full of alternatives, different approaches to every problem. When you’re taking the actual test, choose the approach that you think will be most effective and efficient for any given problem. As long as you have options and remain confident, you will succeed on GRE Quant.