Real GMAT® Problems Ep. 11: Percents, Divisibility, and Quadratics

What This Episode Covers

In Episode 11 of the Real GMAT® Problems series, Isaac works through four real GMAT® problems from the 11th edition of the Official Guide for GMAT® Review: a percent translation question, an algebra question with a fraction denominator, a divisibility question with Roman numerals, and a quadratic with a given root. The common thread is that each problem is decided at the translation step, when the words become written math, and the episode builds the specific translations that make that step reliable.

Isaac also makes the case for memorizing the common fraction-to-percent conversions (thirds, fourths, fifths, eighths, ninths, and elevenths), explains why the parentheses habit when multiplying both sides of an equation is written insurance against distribution slips, and shows how a stated root lets you skip most of the algebra on a quadratic. He skipped one problem from the 11th edition (a coordinate geometry question) because geometry is no longer tested on the GMAT® Focus Edition.

Problems Covered

"What percent of 30 is 12?" A percent translation question with a system behind it: "what" becomes a new variable, "percent" means divide by 100, "of" means multiply, and "is" means equals. Built one word at a time, the question becomes P/100 × 30 = 12, which solves to P = 40. The answer is (D) 40%. Isaac also covers the fraction route (12/30 reduces to 2/5, which is 40%) and which fraction-percent equivalences are worth memorizing.
"If 1.5/(0.2 + x) = 5, then x = ?" An algebra question where the standard move (multiply both sides by the denominator) sets a trap for anyone who forgets to distribute on the right side. The parentheses habit, wrapping both sides of every equation move, makes the distribution nearly impossible to skip. The answer is (B) 0.1; about 7% of test takers pick (D) 0.5 by skipping one distribution.
"If a positive integer n is divisible by both 5 and 7, then n must also be divisible by which of the following?" A Roman numeral question solved by writing divisibility as a fraction (n/(5 × 7) = integer) and checking prime factors. 35 is guaranteed (it's 5 × 7), while 12 and 70 require prime factors the problem never promised. The answer is (C) II only. Testing n = 35 works as a check, but the prime factor system scales where number testing doesn't.
"If 4 is one of the solutions to the equation x² + 3x + k = 10, where k is a constant, what is the other solution?" A quadratic where the given root is a given factor: (x − 4) is already known, and the coefficient on the 3x term finishes the other factor (−4 + 7 = 3), giving (x − 4)(x + 7) = 0. The answer is (A) −7. The full route (solving for k = −18 first, then factoring x² + 3x − 28 = 0) confirms it with more steps.

Key Takeaways

Translate words into math before computing. Every problem in this episode has a step where the given information becomes written structure: a percent sentence becomes an equation, a denominator gets cleared with parentheses, divisibility becomes a fraction, a root becomes a factor. Doing that step in writing, one word or one move at a time, is what keeps questions you know how to do from getting missed.

The four percent translations cover most percent questions. "What" is a new variable (new means new: don't reuse a letter already in the problem), "percent" divides by 100, "of" multiplies, and "is" equals. Memorize them as a set and percent sentences build into equations mechanically, in the order the words arrive.

Memorize the useful fraction-percent equivalences. Thirds, fourths, fifths, eighths, ninths, and elevenths convert percent questions into recognition questions. Sixths and sevenths can wait, since long division covers them. The conversions pay a confidence dividend along with the mathematical one: knowing the material cold feels different in the middle of a section.

Parentheses on both sides of every equation move. When multiplying both sides by a denominator, write parentheses around what you're multiplying by and around what's being multiplied, even when one side obviously doesn't need it. The habit costs nothing and eliminates the distribution slip that 7% of test takers make on the exact question in this episode.

Making mistakes is learning; repeating them is the thing to attack. A mistake made once in practice is part of how anyone learns. A mistake that recurs on questions you already know how to do is the pattern that keeps quant scores below their ceiling, and written process habits are the fix.

Divisibility is a fraction where the denominator must cancel. n divisible by 5 and 7 means n/(5 × 7) produces an integer, which means 5 and 7 appear in n's prime factorization and nothing else is guaranteed. "Must be divisible" questions ask what's guaranteed; "might be divisible" cases aren't selections.

Prime factor analysis scales; number testing doesn't always. Testing n = 35 settles this episode's divisibility question quickly, and testing numbers is a fine default when it works for you. But questions with larger constraints make testing impractical while the prime factor setup barely changes, so it's worth knowing both.

A given root is a given factor. If a quadratic tells you x = 4 is a solution, then (x − 4) is one factor, and the coefficient on the x term finishes the other one. The constant k never matters for that deduction. The plug-and-chug route (solve for k, rebuild, factor from scratch) works and is more concrete, but it spends steps on information the question already gave away.

Fewer steps usually means fewer errors. When two routes both reach the answer, the shorter one tends to be safer, because every step is a chance to slip. Which route to default to is a personal call best settled by review data: solve it both ways when time allows and note which one you'd trust under time pressure.

Real GMAT® Problems Ep. 11: Percents, Divisibility, and Quadratics

What This Episode Covers

Problems Covered

Key Takeaways

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