Demystifying Difficult GRE Math Problems

In the art world, the term trompe l’oeil refers to perspectival illusionism—literally, “to fool the eye.” On the GRE math section, you may notice test questions that use a similar technique in which a readily solvable problem will try to distract you from the very information that makes its solution accessible.
Here’s an example of this tricky sort of test question that appears to involve some rather difficult GRE math:
If x + 2y = 30, then

=

• A. 8
• B. 16
• C. 18
• D. 20
• E. 30

Many test-takers will take one look at this problem and see it more like this:
If x + 2y = 30, then [SCREAMINGLY HORRIBLE, MUST FIND COMMON DENOMINATOR, ARGH!!!] =
Because the latter portion of the problem is so intimidating, many test takers will neglect to notice the answer choices given.

The clever test taker, on the other hand, will force herself to focus on exactly the parts of the test question that are most easy to ignore, first seeing the problem this way:
If x + 2y = 30, then [let’s not worry about finding the common denominator for now] =

• A. 8
• B. 16
• C. 18
• D. 20
• E. 30

Suddenly, by ignoring the intimidating parts of the GRE math, two important ideas come into sharp focus:

1. First, the testmaker is telling you that in order to solve the problem, it matters that x + 2y = 30.
2. Second, despite there being two variables and only one equation, that scary-looking expression can somehow be reduced to a fairly innocent and more manageable integer.

These two observations hold the key to unlocking the solution and avoiding the Test Day panic of rushing to calculate a common denominator head on.
Now, breathe and examine that horrible fractional expression again, looking for x + 2y within it.

In fact, there are two fractions with x as the numerator and two fractions with 2y as the numerator. One of each is over a 3 in the denominator, and the others are over a 5 in the denominator.
It turns out, this is just an addition problem. There’s no reason to have to look at it in the order the test maker gave it to you. Instead, regroup the expression into a more sensible arrangement to simplify the math:

By focusing on the simple information the test maker provides, you can now dispose of this problem easily by replacing each x + 2y with 30:

Making sure that you don’t ignore the subtle signals given by the test makers—rather than being distracted or intimidated by seemingly-impossible GRE math questions—will serve you well on Test Day.

Common GRE Math Mistakes to Avoid

Recognizing a shortcut is only part of solving a difficult GRE math problem. Applying familiar rules accurately matters just as much. Exponents, negative signs, and ratios can lead to mistakes when test takers rush through the calculation.

1. Applying Multiplication Rules to Addition

When multiplying powers with the same base, add the exponents:

2³ × 2⁴ = 2⁷

That rule does not apply when adding powers:

2³ + 2⁴ = 8 + 16 = 24, not 2⁷

For larger expressions, factoring out a common term can make the calculation easier. Consider:

√(2¹⁴ + 2¹⁷)

The square root applies to the entire sum. Since 2¹⁷ = 2¹⁴ × 2³, factor out 2¹⁴:

√[2¹⁴ × (1 + 2³)]
= √(2¹⁴ × 9)
= 2⁷ × 3
= 384

This avoids calculating the large powers individually.

2. Splitting Squares or Square Roots Across Addition

Squaring a sum is different from squaring each term separately:

(a + b)² = a² + 2ab + b²

For example:

(3 + 4)² = 49, but 3² + 4² = 25.

Likewise, a square root cannot generally be split across addition:

√(9 + 16) = √25 = 5

But:

√9 + √16 = 3 + 4 = 7

Testing a questionable rule with small numbers can reveal an invalid shortcut before it causes an error in a more complicated problem.

3. Treating Unusual Exponents as Different Rules

Negative, fractional, or variable exponents may look unfamiliar, but the same exponent rules still apply where the expressions are defined.

For example, raising a power to another power means multiplying the exponents:

(2⁸ˣ)⁻⁰·⁵ = 2⁻⁴ˣ

A negative exponent indicates a reciprocal, not a negative result:

2⁻³ = 1/2³ = 1/8

Writing down the relevant rule before substituting values can help keep the calculation organized.

4. Reversing the Order of Negative Numbers

When comparing negative numbers, the number farther left on a number line is smaller.

For example:

−8 < −5

The same principle applies to negative fractions:

−3/4 < −1/2

Since −3/4 equals −0.75 and −1/2 equals −0.5, −3/4 is farther left. In a Quantitative Comparison problem, checking the signs can prevent an otherwise correct solution from ending with the wrong answer.

5. Misreading Order of Operations and Parentheses

Multiplication and division have equal priority and are performed from left to right. Addition and subtraction also have equal priority and are performed from left to right.

For example:

10 ÷ 5 × 6 = 2 × 6 = 12

Multiplying 5 × 6 first would incorrectly change the expression.

Parentheses also determine whether a negative sign is part of the base being squared:

• (−3)² = 9: The entire negative number is squared.
• −3² = −9: The exponent applies to 3, and the negative sign is applied afterward.

6. Reversing a Ratio

A ratio must follow the order requested in the question. If a group contains 7 designers and 4 engineers:

• The ratio of designers to engineers is 7:4.
• The ratio of engineers to designers is 4:7.

Labeling the quantities on scratch paper helps prevent accidental reversals. Before selecting an answer, test takers should compare the ratio’s order with the wording of the question.

How to Learn from GRE Math Errors

Reviewing a missed question is most useful when it leads to a specific change in approach. Instead of labeling an error “careless,” test takers can record what happened and how to prevent it.

A useful error log includes:

• The mistake: What went wrong, such as reversing a ratio or overlooking a negative sign.
• The correct rule: The mathematical principle needed to solve the problem.
• A prevention step: A concrete habit, such as labeling ratio terms or checking parentheses before applying an exponent.

After reviewing the solution, solving the problem again without looking at the explanation can help confirm understanding. A similar question provides another opportunity to practice the corrected approach.

How to Approach the Hardest GRE Math Problems

Difficult GRE math questions can become more manageable when you change your approach. Along with looking for ways to regroup an expression, try these strategies when a solution isn’t immediately clear:

• Start with a simpler version. Use an easier example to understand the underlying concept, then check what changes in the original problem. In probability questions, for instance, selecting items without replacing them changes what’s available for the next selection.
• Examine the answer choices before calculating. A question involving a large factorial may only require you to identify its factors. Consider whether you can evaluate the choices without calculating the full value.
• Draw the relationships. For coordinate geometry questions, sketch lines or plot sample points to make abstract information easier to interpret. Use the sketch to guide your reasoning, then confirm it mathematically.
• Evaluate every option in select-all-that-apply questions. Check each statement independently. Finding one correct answer doesn’t mean you’re finished; all of the listed options could be correct.