GMAT® Quant: Three Systems for Algebraic Reasoning, Percents, and Word Translations

Most GMAT® quant mistakes don't come from not knowing the math. They come from using the wrong setup or misreading a phrase that changes the entire problem. The computation is correct, but the foundation it's built on was wrong from the start, and the error went unnoticed because the wrong setup looked plausible.

This pattern shows up across three problem types that seem unrelated: algebraic inequalities with fractions, percent calculations with gratuity or tax, and word translations with "in terms of" phrasing. Each one tests a different skill, and each one has a specific trap designed to exploit a common process error. If you approach all three the same way, you'll catch some errors and miss others, because each trap is structurally 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: Algebraic Manipulation for Inequality Problems

When a problem gives you an inequality with variables and asks which expression "must be greater than" or "must be less than" something, the first move is to check whether you can manipulate the given inequality directly into one of the answer choices.

The key insight: if the problem gives you something less than 1 and asks for something greater than 1, there's a good chance you can algebraically rearrange the given information to solve for what's asked. You don't need to invent a new approach. You just need to apply the golden rule of algebra to both sides of the inequality.

For a problem like $\frac{p}{q} < 1$ where $p$ and $q$ are positive integers, multiplying both sides by $q$ gives $p < q$, and dividing both sides by $p$ gives $1 < \frac{q}{p}$. That's the answer, and it took two steps.

Two clues that this approach will work:

1. The problem gives you an inequality and asks for the opposite direction (less than becomes greater than, or vice versa)
2. The answer choices are all recombinations of the same variables

When you see both, algebraic manipulation is almost always the fastest path. Plugging in numbers works as a backup, and logical reasoning can work too, but algebra is the most reliable approach when you can spot the manipulation path.

One caution: when multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign. This problem specifies positive integers, so the sign stays the same. But on problems without that constraint, the sign flip is the most common error.

See this system in action: "If P Divided by Q Is Less Than 1..." — GMAT® Worked Solution

System 2: Set Up Percent Change Correctly

The most common percent error on the GMAT® is treating a final price (including tax, gratuity, or markup) as if it were the original price and simply subtracting the percentage from it.

If a $207 bill includes a 15% gratuity, taking 85% of 207 doesn't recover the original price. Increasing an unknown amount by 15% and setting that equal to 207 is the correct approach. The difference seems subtle, but it produces different answers, and only one is right.

The principle: increasing a number by 15% isn't the same as decreasing the result by 15%. The increased number is larger, so 15% of that larger number is a bigger amount than 15% of the original. Subtracting 15% from the total "undoes" too much, giving you a price that's too low.

The system: set up the equation as $\frac{115}{100} \times P = 207$, where $P$ is the unknown original price. Solve for $P$, then divide by the number of people to get the per-person average.

If setting up the equation feels tricky, the percent change formula is a reliable alternative: $\frac{\text{new} - \text{old}}{\text{old}} \times 100 = \text{percent change}$. Plug in 207 as the new value, $P$ as the old value, and 15 as the percent change. Solve for $P$.

For the computation itself, prime factoring helps when the numbers look unfamiliar. Breaking 207 and 115 into their prime factors makes the cancellation visible and removes the guesswork from long division.

See this system in action: "The Price of Lunch for 15 People Was $207..." — GMAT® Worked Solution

System 3: Translate "In Terms Of" and Eliminate Variables

"In terms of" is one of the most misunderstood phrases on the GMAT®. It means: express the answer using only the variable that follows "in terms of." If a question asks for sedans "in terms of $N$," the answer must contain $N$ and no other variable.

This phrase confuses a lot of test takers, and the confusion makes the problem much harder than it needs to be. Once you understand that "in terms of $N$" means "solve with only $N$ on the other side of the equals sign," the problem becomes a straightforward substitution exercise.

The system for word translation problems with "in terms of":

1. Write what's given and what's asked, clearly and carefully
2. Set up expressions for each quantity in the problem
3. Identify which variable needs to disappear (the one not mentioned in "in terms of")
4. Use substitution to eliminate that variable
5. Solve for the target variable with only the allowed variable remaining

The transcription step is where most errors happen. If the problem says "one fifth of the OTHER cars," writing "one fifth of all cars" changes the entire problem. Double-checking your transcription against the screen takes a few seconds and prevents the most common error on this problem type.

The elimination step is the key inflection point. If you can get an equation with only the target variable and the "in terms of" variable, you're done. Everything else is arithmetic.

See this system in action: "One Third of the Cars Sold Were Sedans..." — GMAT® Worked Solution

Common Mistakes Across All Three Systems

ALGEBRAIC INEQUALITIES: Forgetting to flip the inequality sign when multiplying or dividing by a negative. On problems with positive integers only, this isn't an issue. But when constraints aren't specified, the sign flip is the most common error. The fix: write the constraint at the top of your scratchwork before you start manipulating.

PERCENT CHANGE: Treating the final price as the original and subtracting the percentage from it. This produces an answer that's too low and almost always matches a wrong answer choice. The fix: identify which value is new (the result of the change) and which is old (the original unknown), then set up the equation with the unknown on one side.

WORD TRANSLATIONS: Misreading "other cars" as "all cars" or not understanding what "in terms of" requires. Both errors produce plausible wrong answers that match answer choices. The fix: write what's given and what's asked before doing any math, and double-check your transcription against the screen.

Study Action Items

• For inequality problems, practice spotting when the answer choices are recombinations of the given variables. That's your signal to try algebraic manipulation first.
• Build a flashcard for the percent change formula: (new minus old) divided by old, times 100. Label which value goes where.
• For "in terms of" problems, write the target variable and the allowed variable at the top of your scratchwork. Identify the variable to eliminate before setting up equations.
• Double-check every transcription against the screen after writing it down. Every time, not just when you're unsure.
• Practice prime factoring on unfamiliar numbers. It unlocks cancellation in fraction arithmetic that would otherwise require guesswork.

FAQ

How should I approach GMAT® inequality problems with variables in the answer choices?

Check whether the answer choices are recombinations of the variables in the given inequality. If they are, try algebraic manipulation first. Multiply or divide both sides to rearrange the given inequality into one of the answer choices. If the problem gives you something less than 1 and asks for something greater than 1, there's a good chance you can manipulate directly. Plugging in numbers works as a backup approach, but algebra is usually faster when you can spot the manipulation path.

Why do I keep getting percent problems wrong on the GMAT®?

The most common cause is treating a final price (including tax, gratuity, or markup) as if it were the original price and subtracting the percentage from it. Increasing a number by 15% isn't the same as decreasing the result by 15%, because the increased number is larger and 15% of a larger number is a bigger amount. The fix is to set up the equation with the unknown original price on one side: $\frac{115}{100} \times P = \text{total}$, then solve for $P$.

What does "in terms of" mean on GMAT® word problems?

"In terms of N" means express the answer using only the variable N. The answer choices will contain only N (and numbers), no other variables. To solve these problems, set up equations with all relevant variables, then use substitution to eliminate the variables that aren't allowed in the final answer. If the question asks for sedans "in terms of N," you need an equation with only sedans and N, with no other variables remaining.

How can prime factoring help on GMAT® quant problems?

Prime factoring breaks unfamiliar numbers into their prime components, which makes cancellation visible in fraction arithmetic. When you're dividing or multiplying fractions with large unfamiliar numbers, prime factoring both the numerator and denominator reveals common factors you can cancel. This removes the guesswork from long division and reduces arithmetic errors, especially on problems where the numbers look intimidating but factor cleanly.

Should I use plugging in numbers or algebra on GMAT® algebra problems?

Both approaches work, but they have different risk profiles. Algebra is usually faster when you can spot the manipulation path, but it requires recognizing the opportunity. Plugging in numbers is more reliable but slower, and it can produce multiple matching answers if you pick numbers that are too simple (especially 0 and 1). A good rule: try algebra first when the answer choices are recombinations of the given variables, and fall back to plugging in numbers if the algebra isn't clear within 30 seconds.

Want to Learn Even More?

Listen to Episode 29 of Real GMAT® Problems for the full audio walkthrough of all three problems, including the algebraic manipulation on the inequality problem, the percent change setup on the lunch problem, and the substitution process on the car dealer problem.

For related strategy, read:

• GMAT® Quant: Three Systems for Exponents, Primes, and Percent Change
• GMAT® Quant: Three Systems for Percent, Integer, and Sequence Problems

Worked solutions for this episode:

• "If P Divided by Q Is Less Than 1..." — GMAT® Worked Solution
• "The Price of Lunch for 15 People Was $207..." — GMAT® Worked Solution
• "One Third of the Cars Sold Were Sedans..." — GMAT® Worked Solution