"If the Quotient a/b Is Positive, Which of the Following Must Be True?" — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

If the quotient $\frac{a}{b}$ is positive, which of the following must be true?

(A) $a > 0$

(B) $b > 0$

(C) $ab > 0$

(D) $a - b > 0$

(E) $a + b > 0$

Try it before reading on.

Step 1: Rewrite the Given as an Inequality

"A quotient is positive" becomes $\frac{a}{b} > 0$. Written that way, the given says something specific about the two numbers: a fraction comes out positive when the top and bottom carry the same sign. Either $a$ and $b$ are both positive, or they're both negative.

That reading is the whole problem. The exam calls this subject quantitative reasoning rather than quantitative knowledge, and this is the style it means: the math on the page is a fraction, but the information being delivered is about signs. Two cases, no more, and every answer choice now gets checked against both.

Step 2: Run Every Choice Through Both Cases

"Must be true" means true in every case the given allows, so a choice that fails in one case is finished. One counterexample is enough to retire it.

(A) fails in the both-negative case: $a$ could be $-2$. Choice (B) fails the same way, since $b$ could be $-2$ too.

(C) survives both, and for a reason worth stating: two positive numbers multiply to a positive product, and two negative numbers multiply to a positive product as well. Same signs guarantee a positive product either way.

(D) fails in either case. All the given said was that $\frac{a}{b}$ is positive, not that it's greater than 1, so nothing stops $a = 3$ and $b = 70$: the quotient is still positive, and $a - b = -67$. Even $a = 2$, $b = 2$ breaks it, since $2 - 2 = 0$ isn't greater than zero.

(E) only holds in the both-positive case. With $a$ and $b$ both negative, the sum $a + b$ is definitely not bigger than zero.

The answer is (C).

Step 3: The Same Elimination by Plugging In Numbers

If the same-sign reading doesn't click, plugging in numbers gets there too, and it's a legitimate route rather than a consolation prize. The clue is in the problem itself: whenever "greater than 0," "less than 0," "positive," or "negative" shows up, the question is almost always testing positives versus negatives, so the numbers to pick are one pair of positives and one pair of negatives.

Take $a = 2$, $b = 2$. The constraint allows it, since $\frac{2}{2} = 1$ is positive. Testing the choices, only (D) fails, since $2 - 2 = 0$. One choice down, and traction established.

Then take $a = -2$, $b = -2$, which the constraint also allows, since $\frac{-2}{-2} = 1$ is still positive. This time (A), (B), and (E) all fail, leaving (C) standing.

The tradeoff is time: plugging in usually takes longer than the theory, and it's often the safer route, especially when the theoretical approach feels shaky or the pressure is high. Whichever route a person favors, keeping the other one ready is how the quant section gets resilient, because some problems are written to trip up one approach or the other. For score targets above the mid-90s percentiles, doing problems more than one way starts to pay dividends.

Why This Problem Matters

The trap in choices (A) and (B) is answering "can be true" when the question asked "must be true." Statement (A) is true often, and often isn't the bar: the both-negative case kills it, and the question survives only if a statement survives every case.

The same-sign pattern is worth a note in a memorization system, because $\frac{a}{b} > 0$ is a common way of saying "these two numbers share a sign" without saying it. The construction can feel unfamiliar on a first encounter, and that's normal; pattern recognition comes from repeated exposure.

Ready for the next one? "How Many Minutes Does It Take John to Type Y Words..." — GMAT® Worked Solution.

Want the full strategy? Read: GMAT® Quant Solution Paths: Theory, Plugging In, and Brute Force

From Episode 9 of Real GMAT® Problems (The GMAT® Strategy Podcast).