GMAT® Quant: Three Systems for Exponents, Primes, and Percent Change
Most GMAT® quant mistakes don't come from not knowing the math. They come from setting up the problem incorrectly and not catching the error before time runs out. The computation is right, but the setup was wrong, and the error went unnoticed because the setup looked correct.
This pattern shows up across three problem types that seem unrelated: exponent inequalities, prime number properties, and percent change. Each one tests a different skill, and each one has a specific trap designed to exploit a common setup error. If you use the same approach for all three, you'll catch some errors and miss others, because each trap is different.
The fix isn't "be more careful." The fix is having a specific system for each problem type, so the system catches the trap before it costs you points.
System 1: Unify the Bases for Exponent Inequalities
When an exponent problem gives you different bases on each side of an equation or inequality, the first move is almost always to rewrite the bases so they match. If you can get the same base on both sides, you can ignore the bases and just compare the exponents.
For example, if you see , rewrite 25 as and simplify using the power-of-a-power rule: . The inequality becomes , which means , so . The bases matched, the exponents compared directly, and the algebra was straightforward.
The trap on these problems isn't the exponent rule. It's transcribing the inequality as an equation. You do all the math correctly, but somewhere between reading the problem and writing it on your scratch pad, the "" becomes "". You get instead of , and you pick 6 instead of 7. The difference is one character on your scratch pad, and it's the difference between the right answer and the most popular wrong answer.
The system: write the question down before you start solving, and double-check your transcription against the screen before you proceed. Two habits that take seconds and prevent the most common error on this problem type.
System 2: Must-Be-True Elimination for Prime Number Properties
"Must be true" questions with Roman numerals are built for elimination. You don't need to evaluate all three statements at once. You need to find the easiest one, prove or disprove it, and eliminate answer choices immediately.
The system:
- Scan all three Roman numerals and pick the easiest one to evaluate first
- Prove or disprove it
- Eliminate any answer choice that doesn't include or excludes that statement
- Move to the next easiest statement
- Repeat until one answer remains
The "easiest" statement is usually the one that requires the least computation. If one statement is about even/odd properties and another requires finding large prime numbers, start with the even/odd one.
For prime number problems specifically, the key insight is that all primes greater than 2 are odd. That single fact resolves most prime number property questions on the GMAT®. The sum of two odd numbers is even. The difference of two odd numbers is even. The quotient of two different primes is never an integer.
The trap on these problems is forgetting constraints. If the problem says "greater than 2," you need that constraint written on your scratch pad. Without it, you might start thinking about 2 as a prime number, and 2 plus 89 equals 91, which would make a "must be true" statement seem false. The constraint changes the entire problem.
When a statement seems hard to evaluate, try listing the smallest possible values systematically. Start with the smallest prime that satisfies the constraints, pair it with the next smallest, and look for a pattern. You almost never need to list more than 4 or 5 pairs before the pattern becomes clear.
System 3: Percent Change Formula for Price Reductions
The percent change formula is the most reliable system for price reduction problems:
The formula does two things that prevent errors. First, it forces you to identify which value is new and which is old before you do any arithmetic. Second, it handles the direction of change automatically: a decrease gives a negative result, which you can convert to a positive "reduction" percentage.
The trap on these problems is almost always in the setup, not the computation. Two errors account for most wrong answers:
- Swapping new and old values, which produces a slightly different percentage
- Computing for the wrong quantity, like one item instead of two
The fix: write what's given and what's asked before you plug anything into the formula. If the problem asks about two candy bars, your old value is the regular price of two candy bars, not one. If the problem says "sale price," the sale price is the new value, not the old one.
A negative result from the formula means a decrease, which is expected for a "reduction" problem. The answer choices will be positive percentages, so drop the negative sign when the problem asks for a "reduction" or "decrease."
Common Mistakes Across All Three Systems
EXPONENTS: Solving as an equation when the problem gives an inequality. The math is identical, but the answer changes from 6 to 7. The fix: write the inequality symbol on your scratch pad and double-check it before selecting your answer.
PRIMES: Forgetting the constraint that primes are greater than 2. Without that constraint, 2 is a valid prime, and statements that should always be true suddenly have exceptions. The fix: write all constraints at the top of your scratchwork before evaluating any statement.
PERCENT CHANGE: Computing for the wrong quantity or swapping old and new values. Both errors produce a plausible-looking answer that matches a wrong answer choice. The fix: write what's given and what's asked, label each value as "old" or "new" before plugging into the formula.
Study Action Items
- Practice base unification with 5-10 exponent problems. Focus on rewriting bases to match, not on the arithmetic.
- Build a flashcard for the percent change formula. Label which value goes where.
- Memorize: all primes greater than 2 are odd. This single fact resolves most prime number property questions.
- For every problem, write what's asked and all constraints at the top of your scratchwork before doing any math.
- Double-check your transcription against the screen after writing it down, every time.
FAQ
How should I set up GMAT® exponent inequality problems?
Rewrite all bases to match, then compare exponents directly. If both sides have the same base (and the base is greater than 1), you can ignore the bases and just work with the exponents. Always transcribe the inequality symbol carefully, since solving as an equation instead of an inequality is the most common error on these problems.
Why do I keep getting "must be true" questions wrong?
The most likely cause is forgetting a constraint from the problem. If the problem specifies that primes are "greater than 2," you need that written on your scratch pad. Without it, you might consider 2 as a prime, which creates exceptions to statements that should always be true. Write all constraints before evaluating any Roman numeral statements, and use systematic listing to find patterns when a statement is hard to evaluate directly.
What is the percent change formula and when should I use it?
The formula is (new minus old) divided by old, times 100. Use it whenever a problem asks for a percent increase, percent decrease, or percent reduction. Identify which value is new (the result of the change) and which is old (the original value) before plugging in. A negative result means a decrease, which you express as a positive percentage when the problem asks for a "reduction."
How much time should I spend on exponents during GMAT® prep?
Exponent rules, including base unification and power-of-a-power, are worth prioritizing early because they appear in many different problem types. If you can rewrite bases to match and apply exponent rules reliably, you'll have the foundation for a significant portion of GMAT® quant.
Should I use the answer choices on GMAT® exponent problems?
Yes. Plugging in the answer choices and testing them against the inequality is a valid strategy, especially when the algebra is complex or you're unsure of your setup. For problems with integer answer choices and exponents, testing values can be faster than doing the algebra.
Want to Learn Even More?
Listen to Episode 27 of Real GMAT® Problems for the full audio walkthrough of all three problems, including the base unification on the exponent inequality, the elimination process on the prime number properties question, and the percent change formula setup on the candy bar problem.
For related strategy, read:
- GMAT® Quant: Three Systems for Algebra, Compound Interest, and Estimation
- GMAT® Quant: Three Systems for Percent, Integer, and Sequence Problems
Worked solutions for this episode: