GMAT® Percents, Divisibility, and Quadratics: Translate First, Solve Second

Percents, divisibility, and quadratics tend to live in three separate mental drawers. A percent question feels like a vocabulary problem, divisibility feels like a memory problem, and quadratics feel like a factoring problem, so preparing for all three can feel like three separate review projects.

Episode 11 of our Real GMAT® Problems series works through four Official Guide problems that cut across all three drawers, and what the solutions share is more useful than what the topics don't. Each one is decided at the translation step, the moment the question stops being words and starts being written math. Get the translation written down correctly and the computation that follows is usually the easy part.

Percent Words Have Fixed Translations

Four words carry almost every percent translation question:

They apply in the order the words appear. "What percent of 30 is 12?" builds left to right into $\frac{P}{100} \times 30 = 12$, with no percent intuition needed at any point during the build. The computation then takes over: multiply both sides by 100, divide by 30, and $P = 40$.

Two details keep the system reliable. "What" has to become a variable that isn't already in use, because reusing $x$ for two jobs is where translations start contradicting themselves. And "percent" attaches to whatever word came before it, so $P$% arrives as $\frac{P}{100}$ and 20% arrives as $\frac{20}{100}$, ready for whatever multiplication or equals sign comes next.

For test takers who already convert $\frac{12}{30}$ to $\frac{2}{5}$ to 40% on sight, the system can feel like training wheels. It earns its spot on the questions where the first move stalls, and those tend to be the higher-difficulty ones.

See this system in action: "What Percent of 30 Is 12?" — GMAT® Worked Solution

Clear Denominators With Parentheses on Both Sides

An equation with a complicated denominator, like $\frac{1.5}{0.2 + x} = 5$, usually wants the denominator cleared first, and multiplying both sides by the full denominator is the standard move.

The habit that protects the move is writing a parenthesis on each side: $(1.5) = (5)(0.2 + x)$. The parentheses cost two keystrokes and buy a near-zero chance of skipping the distribution, because $(5)(0.2 + x)$ visibly contains two terms waiting to be multiplied. Written without them, the same line invites the version where the 5 multiplies the 0.2 and forgets the $x$, which produces $1.5 = 1 + x$, which produces $x = 0.5$, which is a stocked wrong answer on the actual question. About 7% of test takers take that trip.

The deeper pattern is that questions you know how to do can still get missed, and process is what protects them. Parentheses on both sides is a written defense: it works even when attention is spent, which is the state most of us are in an hour into the quant section.

See this system in action: "If 1.5/(0.2 + x) = 5, Then x = ?" — GMAT® Worked Solution

"Divisible" Means the Denominator Cancels

Divisible has a precise meaning: when $x$ is divisible by $y$, the fraction $\frac{x}{y}$ produces an integer. Written that way, divisibility questions become cancellation questions, because the fraction comes out even exactly when every prime factor of the denominator cancels with something in the numerator.

The question "if $n$ is divisible by both 5 and 7, then $n$ must also be divisible by which of the following?" starts from $\frac{n}{5 \times 7} = \text{integer}$, which fixes everything known about $n$: its prime factorization contains a 5 and a 7, and nothing else is guaranteed. Roman numeral II, which is 35, is $5 \times 7$ prime factored, so it cancels in every allowed case. Roman numeral III, which is 70, is $2 \times 5 \times 7$, and the 2 might not exist in $n$. Roman numeral I, which is 12, has no 5 or 7 at all. Only II must divide $n$.

The must-versus-might distinction is where the points live on these questions. A numeral that divides $n$ in the cases you happen to picture is a might, and a might isn't an answer. Prime factors turn that judgment call into a checkable list.

See this system in action: "If a Positive Integer n Is Divisible by Both 5 and 7..." — GMAT® Worked Solution

A Given Root Is a Given Factor

When a quadratic hands you one solution, it has handed you one factor, free. On the question "if 4 is one of the solutions to $x^2 + 3x + k = 10$, what is the other solution?", the root $x = 4$ means $(x - 4)$ is one binomial of the factored form.

The other binomial comes from the coefficient on the $3x$ term: the two constants in the factors must sum to 3, so $-4$ pairs with 7, the factored form is $(x - 4)(x + 7) = 0$, and the other solution is $-7$. The constant $k$ never enters the reasoning.

The full route, solving for $k$ first ($k = -18$), rebuilding the equation ($x^2 + 3x - 28 = 0$), and factoring from scratch, reaches the same answer with more steps. Which route to prefer isn't a universal; it's a personal call that review data settles. But the shorter route exists only for test takers who notice what the given already buys, and that noticing is a habit worth building everywhere on the quant section, because questions leave gifts like this constantly.

See this system in action: "If 4 Is One of the Solutions to x² + 3x + k = 10..." — GMAT® Worked Solution

Common Mistakes

Answering the flipped percent question. "What percent of 30 is 12?" stocks 250% and 2.5% for anyone who computes $\frac{30}{12}$ instead of $\frac{12}{30}$. The word-by-word translation can't flip, because it builds in the order the words arrive.

Distributing to one term instead of both. The 7% who pick 0.5 on the fraction question aren't short on algebra knowledge; they skipped one distribution on one line. Parentheses on both sides of every equation move is the written fix.

Answering "might" when asked "must." On divisibility questions, a factor that divides $n$ in some allowed cases isn't a valid selection. The cancellation setup makes the requirement concrete: every prime factor of the chosen numeral has to be guaranteed by the stem.

Re-deriving what was given. A stated root is a stated factor. Solving for $k$ and factoring from scratch works, but it spends time and adds steps to a question that was already half-factored on arrival.

Study Action Items

• Make a flashcard for the four percent translations (what, percent, of, is) and drill it until the equation builds without deliberation.
• Adopt the two-parentheses habit for every equation move, on easy questions first, so it's automatic on hard ones.
• Write divisibility statements as fractions with "= integer" and check prime factors instead of dividing.
• When a quadratic gives you one root, write the matching factor before reaching for the algebra.
• During review, solve questions with two viable routes both ways, and record which one you'd trust under time pressure.

Problems From This Episode

• "What Percent of 30 Is 12?" — GMAT® Worked Solution
• "If 1.5/(0.2 + x) = 5, Then x = ?" — GMAT® Worked Solution
• "If a Positive Integer n Is Divisible by Both 5 and 7..." — GMAT® Worked Solution
• "If 4 Is One of the Solutions to x² + 3x + k = 10..." — GMAT® Worked Solution

FAQ

How do I translate a percent word problem on the GMAT®?

Four words do the work: "what" becomes a new variable, "percent" means divide by 100, "of" means multiply, and "is" means equals. Apply them in the order the words appear, and "What percent of 30 is 12?" becomes $\frac{P}{100} \times 30 = 12$ before any arithmetic starts. Then solve the equation with ordinary algebra.

What does "divisible" mean in GMAT® questions?

It means the division produces an integer with no remainder. Written as a fraction, $x$ divisible by $y$ requires every prime factor of $y$ to cancel with a prime factor of $x$. That framing turns "must be divisible by" questions into prime-factor checks instead of division practice.

How do I find the other root of a quadratic when one root is given?

A given root is a given factor. If 4 is a solution, then $(x - 4)$ is one factor, and the two constants in the factors must sum to the coefficient of the $x$ term. Solve for the missing constant, and the second factor gives you the other root without ever needing the constant $k$.

Should I memorize fraction-to-percent conversions for the GMAT®?

The thirds, fourths, fifths, eighths, ninths, and elevenths are worth memorizing, because they convert percent questions into recognition questions. Sixths and sevenths can be skipped, since long division covers them when they appear. The memorized conversions also tend to pay a confidence dividend in the middle of the exam, which is its own kind of return.

Want to Learn Even More?

Listen to Episode 11 of Real GMAT® Problems from The GMAT® Strategy Podcast for the full discussion, including why memorized fraction equivalences earn their keep and when number testing stops scaling.

Related reading:

• Translating Percent Word Problems on the GMAT®: A One-Word-at-a-Time System
• GMAT® Ratios: How to Translate Word Problems Into Algebra
• GMAT® Quant Fundamentals: Process Over Computation