GMAT® Quant Solution Paths: Theory, Plugging In, and Brute Force

One solution per problem feels efficient: work the theory, pick the answer, move on. Then a question arrives whose theory never quite clicked, or whose arithmetic runs four lines long, and the single road is blocked with the clock running. Episode 9 of our Real GMAT® Problems series works through three Official Guide problems, and the thread running through all three is that every question offered more than one road: theory or plugging in numbers on the first, algebra or a rate chart on the second, brute force, prime factoring, or perfect-square factoring on the third.

The roads matter because the biggest score killer in the quant section probably isn't lack of speed. It's missing questions you know how to do. A solution path you trust under pressure is worth more than a faster path you might slip on, and each problem below shows what that looks like in practice.

Reading "Must Be True" as a Two-Case Problem

Questions that ask which statement "must be true" are asking which statement holds in every case the given information allows. The given on this episode's first problem is $\frac{a}{b} > 0$, and the fraction is doing sneaky work: a quotient comes out positive when the top and bottom carry the same sign, so the given is really a statement about $a$ and $b$ sharing a sign. Both positive, or both negative. Two cases, no more.

That reading turns the answer choices into a two-column check, and one counterexample is enough to eliminate any choice that fails in a single case. It's also a pattern worth recognizing on sight, because the same-sign construction reappears on harder questions wearing fancier clothing. The full walkthrough is in "If the Quotient a/b Is Positive, Which of the Following Must Be True?" — GMAT® Worked Solution, including the numbers-first version for when the theory doesn't click.

Rows and Columns: The Word Problem Setup That Scales

Word problems are organizational challenges more often than mathematical ones, and the setup that handles most of them is a spreadsheet minus the lines: the formula across the top row, one new row per instance of that formula in the problem. On a rate question, the header is Rate × Time = Work, units written in every cell.

The units are not decoration. Words per minute times minutes leaves words, and watching that happen on the page is a triple check that the logic hasn't been reversed, because reversing it is exactly what happens under time pressure. The empty cell tends to be the quantity the question wants, which means the chart points at the answer before any algebra starts. Investing in the setup can feel slower while writing it, and it can save time by the end of the problem.

See this setup in action: "How Many Minutes Does It Take John to Type Y Words..." — GMAT® Worked Solution

Three Roads Through a Big Square Root

The third problem asks for $\sqrt{16 \times 20 + 8 \times 32}$, and the arithmetic inside the root is manageable: $16 \times 20 = 320$, $8 \times 32 = 256$, and the sum is 576. Then the road forks. One route is recognition: 576 is $24^2$, and the answer drops out. A second route is prime factoring, which turns 576 into $2^6 \times 3^2$ and takes the square root by cutting every exponent in half. A third is factoring a perfect square out of both terms, which converts the addition under the root into multiplication, the one operation roots do distribute over.

None of the three requires the others, and that's the point. Memorization gaps can't stall a problem that prime factoring can finish, and missing insight can't stall a problem that brute force can finish. The full comparison is in "√(16 × 20 + 8 × 32) = ?" — GMAT® Worked Solution.

Common Mistakes

Selecting a "might" on a "must" question. Choice (A) on the quotient problem is often true, and often isn't enough. "Must" means every allowed case, so $a > 0$ goes out the moment one negative pair works, and testing both signs is the habit that catches it.

Dropping the units on a rate setup. The $XY$ trap on the typing problem catches people who skip the chart and multiply the two givens, because the problem looks too simple to need one. Units written next to every variable expose the reversal before it costs the question.

Distributing a root over addition. $\sqrt{a + b}$ doesn't split into $\sqrt{a} + \sqrt{b}$. Roots distribute over multiplication only, so the sum under the radical needs a perfect square factored out first, which converts the addition into multiplication.

Treating speed as the starting point. Speed comes from repetition, and speed built on sloppy habits is progress toward a score nobody wants. Start good and slow, then add pressure once the habits hold.

Study Action Items

• When "positive," "negative," "greater than 0," or "less than 0" appears in a problem, plan to test both signs before reading the choices.
• Set up every rate word problem as rows and columns with units in every cell, starting on the easy ones, where the stakes are low enough for the habit to form.
• Memorize the perfect squares through $20^2$, and let prime factoring cover whatever memory drops.
• During review, solve a problem you got right a second way, and note which route you would trust under time pressure.

Problems From This Episode

• "If the Quotient a/b Is Positive, Which of the Following Must Be True?" — GMAT® Worked Solution
• "How Many Minutes Does It Take John to Type Y Words..." — GMAT® Worked Solution
• "√(16 × 20 + 8 × 32) = ?" — GMAT® Worked Solution

FAQ

What does "must be true" mean on a GMAT® quant question?

It means true in every case the given information allows, not just sometimes true. The method is to list the allowed cases and eliminate any choice that fails in one of them, since a single counterexample is proof that a statement doesn't have to hold. On the quotient question, $\frac{a}{b} > 0$ allows two cases (both numbers positive, both negative), and only $ab > 0$ survives both.

What does a positive quotient tell you about the signs of two numbers?

That the two numbers share a sign: both positive or both negative. A fraction is positive when the numerator and denominator match, so $\frac{a}{b} > 0$ can hold with $a$ and $b$ both negative, which means neither number has to be positive on its own. It's a common construction, a sign relationship delivered in the language of fractions.

How should I set up a GMAT® rate word problem?

As rows and columns: the formula (Rate × Time = Work) across the top, one row per instance of the formula, units written in every cell. On the typing problem, $X$ words per minute $\times$ $t$ minutes = $Y$ words, and the units confirm the setup before any solving starts. Dividing both sides by $X$ gives $t = \frac{Y}{X}$.

How do I simplify a square root I haven't memorized, like √576?

Prime factor the number and halve every exponent. Since $576 = 2^6 \times 3^2$, the square root is $2^3 \times 3^1 = 24$. Memorizing perfect squares through $20^2$ makes the recognition route faster, but prime factoring covers any number memory drops.

Can you distribute a square root over addition?

No. Roots distribute over multiplication, so $\sqrt{16} \times \sqrt{36} = \sqrt{576}$, but $\sqrt{a + b}$ doesn't split into $\sqrt{a} + \sqrt{b}$. The workaround on problems with a sum under the radical is factoring a perfect square out of both terms first: $\sqrt{16 \times 20 + 16 \times 16} = \sqrt{16(20 + 16)} = \sqrt{16 \times 36}$, and then the root distributes.

Want to Learn Even More?

Listen to Episode 9 of Real GMAT® Problems from The GMAT® Strategy Podcast for the full discussion, including the case for plugging in numbers as a default rather than a fallback, and the argument that the safest route beats the fastest one.

Related reading:

• GMAT® Word Problems: When Picking Numbers Beats Algebra
• GMAT® Exponents and Roots: A Complete Guide
• Be Accurate to Improve Your GMAT® Score: Why Precision Comes Before Speed