"If P Divided by Q Is Less Than 1..." — GMAT® Worked Solution

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

If $\frac{p}{q} < 1$ and $p$ and $q$ are positive integers, which of the following must be greater than 1?

(A) $\sqrt{\dfrac{p}{q}}$

(B) $\dfrac{p}{q^2}$

(C) $\dfrac{p}{2q}$

(D) $\dfrac{q}{p^2}$

(E) $\dfrac{q}{p}$

Try it before reading on.

Approach 1: Algebraic Manipulation

The problem gives us $\frac{p}{q} < 1$ and asks which expression must be greater than 1. The answer choices are all recombinations of $p$ and $q$, which is a strong signal that algebraic manipulation will work.

Start with the given inequality:

$\frac{p}{q} < 1$

Since $p$ and $q$ are positive integers, we can multiply both sides by $q$ without flipping the inequality sign:

$p < q$

Now divide both sides by $p$ (again positive, so no sign flip):

$1 < \frac{q}{p}$

That's option (E). Two steps, no computation.

How to Spot This Approach

Two clues make algebraic manipulation the natural choice here:

First, the problem gives you something less than 1 and asks for something greater than 1. When the target is the "opposite direction" of what's given, you can often rearrange the given information to get there.

Second, the answer choices are all different recombinations of the same two variables. When you see $p$ and $q$ arranged in fractions, squared, under roots, and swapped across numerator and denominator, that's a signal that the problem is testing whether you can rearrange the given relationship, not whether you can compute specific values.

Approach 2: Plugging In Numbers

If the algebraic path isn't obvious, plugging in numbers works. We need $p$ and $q$ to be positive integers where $\frac{p}{q} < 1$, meaning $p < q$.

Let's pick $p = 3$ and $q = 5$. Check the constraint: $\frac{3}{5} = 0.6 < 1$. Good.

Now evaluate each answer choice:

• (A) $\sqrt{\frac{3}{5}} = \sqrt{0.6} \approx 0.77$ (less than 1)
• (B) $\frac{3}{25} = 0.12$ (less than 1)
• (C) $\frac{3}{10} = 0.3$ (less than 1)
• (D) $\frac{5}{9} \approx 0.56$ (less than 1)
• (E) $\frac{5}{3} \approx 1.67$ (greater than 1)

Only (E) is greater than 1. If you happen to pick numbers that produce multiple answers greater than 1 (which is rare), choose new numbers that still satisfy the constraints and test again. Stay away from 0 and 1 when picking numbers, since they can produce ambiguous results.

Approach 3: Logical Reasoning

If $\frac{p}{q} < 1$ with positive integers, then $p < q$. The fraction $\frac{p}{q}$ is less than 1 because the numerator is smaller than the denominator.

Flipping that fraction to $\frac{q}{p}$ swaps the roles: the numerator is now larger than the denominator. Any fraction where the numerator exceeds the denominator (with positive values) is greater than 1.

Think of it with concrete numbers. $\frac{5}{6} < 1$. Flip it: $\frac{6}{5} > 1$. $\frac{10}{11} < 1$. Flip it: $\frac{11}{10} > 1$. The pattern holds for any positive integers where $p < q$.

This reasoning is fast and elegant, but it requires practice to use reliably under pressure. If logical reasoning comes naturally to you, it's a valid tool. If it doesn't, the algebraic approach gives the same result with a more structured process that's harder to fumble.

The answer is (E).

Why This Problem Matters

This is a warm-up problem that tests basic algebraic reasoning, but it has a few pitfalls. The most common one isn't the math itself, it's failing to recognize that multiple approaches exist and wasting time on a harder path when a simpler one is available.

The broader lesson: when answer choices are recombinations of the given variables, look for a direct algebraic manipulation first. It's usually the fastest and most reliable approach. Plugging in numbers works as a backup. Logical reasoning works when you've practiced it enough to trust your instincts, but it can backfire if you apply it carelessly on harder problems.

The same pattern appears across many GMAT® quant problems: the "fast" approach looks appealing, but the "safe" approach (algebra, plugging in numbers) has a higher success rate across the full range of difficulty. Using fast approaches on easy problems and safe approaches on harder ones is a reasonable strategy, but you need to recognize which is which.

Want the full strategy behind this problem? Read: GMAT® Quant: Three Systems for Algebraic Reasoning, Percents, and Word Translations

From Episode 29 of Real GMAT® Problems (The GMAT® Strategy Podcast).

Ready for the next one? "The Price of Lunch for 15 People Was $207..." — GMAT® Worked Solution.