The GRE Quantitative section doesn't just test your math knowledge. Above all, it assesses your problem-solving skills. ETS, maker of the exam, suggests some general steps to follow when solving quantitative problems. While a fairly new offering from the test maker, the three-step approach that ETS describes is based on a classic set of problem solving principles.

### ETS's General Problem Solving Steps

In the Official GRE Quantitative Reasoning Practice Questions, published in 2014, ETS proposes that a math problem can be solved in three basic steps:

Although ETS cites no sources, the test maker is adapting the work of mathematician George Polya (d. 1987). Highly regarded among math educators (including The National Council of Teachers of Mathematics), Polya divided the problem-solving process into four phases:

Each phase follows a basic principle of problem solving. ETS combines *Devising* and *Carrying Out* in a single step and reduces *Looking Back* to checking your answer rather than, as Polya encouraged, reflecting on your solution—an activity that's important when practicing but impractical when testing.

### Problem Solving in Action

No one gets good at problem solving by just talking about it. You have to spend lots of time doing it. So here's a problem for practice.

Two trains depart different terminals at the same time and travel toward each other. Train S moves at a constant rate of 60 miles per hour, and train T moves at a constant rate of 90 miles per hour. If the two trains start 450 miles apart and ride straight parallel tracks over flat terrain, how many miles will train S have traveled when it meets train T?

- 180
- 225
- 270
- 300
- 325

Ready for the Polya-style breakdown? Here's how the four phases of solving this problem might shake out.

#### Understanding the problem

You're challenged to find an unknown. What's unknown is a distance traveled, and what's given are constant rates, along with the condition that the travelers leave at the same time from opposite ends of parallel paths that are equal in length.

Sounds like you have some algebra to do. More precisely, you have a distance word problem to solve. You also have a list of distances (the answer choices), one of which must be the unknown. That list may come in handy.

#### Devising a plan

There are at least a couple different ways to approach this problem. One way is to **Translate Words to Algebra**, **Use a Formula**—namely, distance = rate × time—and **Solve an Equation**. Polya wrote about each of these strategies, and ETS talks about all three, too. You probably practiced these sorts of mathematical maneuvers a lot in junior high and high school.

Another approach is to **Solve a Simpler Problem and Adapt the Solution**. (ETS and Polya would likely lump this strategy in with Adapting Solutions to Related Problems.) You can also **Analyze the Answer Choices** to try to exploit the multiple-choice format. We'll do both: we'll adapt the solution from a simpler problem and then we'll analyze the answer choices. We'll also throw in a little translation.

#### Carrying out the plan

Since S and T leave at the same time, they'll have been moving the same amount of time when they meet each other. They'll also have gone a combined 450 miles, since they start 450 miles apart.

- S's time = T's time
- S's distance + T's distance = 450

Imagine S and T are traveling at equal constant rates. When the two trains meet, S and T will have traveled equal parts of 450 miles— 225 miles each. So the solution to this simpler, equal rates problem would be choice (B).

- Same times, Same rates → Same distances → 225

Now adapt to the problem at hand. If equal rates mean equal distances, then unequal rates mean unequal distances.

- Same times, Different rates → Different distances

The faster train, T, will go further than the slower train, S. Since their combined distance when they meet is 450, S's distance can't equal or exceed 225; otherwise, it would equal or exceed T's distance. So S went less than 225 miles.

- S's distance < 225 < T's distance

Looking at the options, only (A) is less than 225. So (A) must be correct.

#### Looking back

In this final phase, you want to at least follow ETS's advice and check your answer. Here are some questions to ask yourself.

- Did you use all the necessary info and stick to the condition in the problem?
- Does your answer fit the problem? For instance, does it take the right form—integer, decimal, variable, expression, etc? If your answer is a value, does it seem reasonable, rather than too big or too small (potentially excluding it from the available answers, if multiple choice)?
- Did you avoid calculation errors and other easily overlooked mistakes?

Everything checks out. Interestingly, calculation errors don't really threaten our solution, since it's more about higher-level quantitative reasoning (which GRE Quant aims to measure) than lower-level calculations. And that's worth reflecting on.

Want more problem solving practice? Try the problems in these posts:

- GRE Data Analysis Challenge: Compare Standard Deviations
- GRE Algebra Challenge: Quadratics & Negative Exponents
- GRE Geometry Challenge: Use the Volume Formula Wisely
- GRE Arithmetic Challenge: Find the Expression that Must Be Negative
- GRE Word Problem Challenge: Extreme Meal Pricing
- GRE Data Analysis Challenge: Probability at Maximum Difficulty