"If x and y Are Positive, Which Must Be Greater Than 1/√(x + y)?" — GMAT® Worked Solution

From Episode 45 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind plugging in numbers, read: GMAT® Algebra: What to Do When You're Stuck.

The Problem

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

If x and y are positive, which of the following must be greater than 1 / √(x + y)?

I. √(x + y) / (2x)
II. (√x + √y) / (x + y)
III. (√x − √y) / (x + y)

(A) None
(B) I only
(C) II only
(D) I and III
(E) II and III

Try it before reading on.

Why This Problem Is Hard

This is effectively three problems in one. You need to evaluate each Roman numeral separately and get all three right for the overall answer to be correct.

The algebra here involves comparing square root expressions to a fraction with a square root in the denominator. You can do it, but there are a lot of places to stumble — and the form-matching challenge from the earlier problems in this episode gets even worse with square roots involved.

Students who plug in numbers on this problem tend to show about 40% higher accuracy and finish about 60% faster than those who use pure algebra. That's a big gap.

Picking Good Numbers

You want numbers that make the computation as simple as possible.

The original expression has √(x + y) in the denominator. If x + y gives you a perfect square, the square root becomes a clean integer. That makes everything easier.

Try x = 2 and y = 2.

x + y = 4. √4 = 2. The original expression becomes 1/2.

So the question becomes: which Roman numerals give a value greater than 1/2?

Write it down: x = 2, y = 2, target = 1/2.

Testing Roman Numeral I

√(x + y) / (2x) = √4 / (2 × 2) = 2/4 = 1/2.

1/2 is NOT greater than 1/2. It's equal.

So Roman numeral I does not HAVE to be greater. It might be sometimes, but "must be greater" means 100% of the time. One counterexample is enough.

Eliminate every answer choice that includes I. That knocks out (B) and (D).

Three answer choices left: (A), (C), and (E).

Testing Roman Numeral III

(√x − √y) / (x + y) = (√2 − √2) / (2 + 2) = 0/4 = 0.

Zero is less than 1/2. So Roman numeral III does not have to be greater either.

Eliminate every answer choice that includes III. That knocks out (E).

Two answer choices left: (A) and (C).

Testing Roman Numeral II

(√x + √y) / (x + y) = (√2 + √2) / (2 + 2) = 2√2 / 4.

The 2 in the numerator cancels with the 4 in the denominator: √2 / 2.

Now you need to figure out if √2 / 2 is greater than 1/2.

If you've memorized that √2 is roughly 1.4, this is quick: 1.4/2 = 0.7, which is greater than 0.5.

Even without that approximation, you can reason it out. √1 = 1, so √2 has to be bigger than 1. That makes √2/2 bigger than 1/2.

Roman numeral II is greater than the original expression for this set of values.

Choosing Between (A) and (C)

You're down to two options:

(A) says none of the Roman numerals are always greater. (C) says II is always greater.

You've shown that II CAN be greater. So the question is whether it MUST be greater — every time, for all valid values of x and y.

At this point, you have a few options:

Test another set of numbers to build confidence. Try x = 3 and y = 1. Target: 1/√4 = 1/2. Roman numeral II: (√3 + 1)/4. √3 is about 1.73, so the numerator is about 2.73. 2.73/4 is about 0.68. Still greater than 0.5.

Try algebra on just Roman numeral II. That's faster than doing algebra on all three — and you only have one thing to prove or disprove.

Make an educated bet. You've eliminated three answer choices. Your odds went from 20% to 50%. If you're short on time, that's a good return.

In practice, students who plug in numbers on this problem almost always land on the right answer. The uncertainty is real, but the accuracy is high.

The answer is (C).

What to Take Away

Roman numeral questions are ideal for plugging in numbers. You evaluate each Roman numeral independently. When one fails for your test values, you eliminate every answer choice that includes it. One round of testing often cuts your options from five down to two or three.

Pick numbers that create perfect squares under square roots. On this problem, x = 2 and y = 2 made the original expression a clean 1/2. That's the difference between quick mental math and a messy computation with decimals.

The "none" option creates a unique challenge. When "none" is one of the remaining choices, you can't be 100% certain from plugging in alone. But you can build confidence with multiple test values, use algebra on the one remaining Roman numeral, or accept the improved odds. All of those are better than being stuck with no approach at all.

Having a backup approach changes the game. The algebra on this problem is difficult. But plugging in numbers with a good choice of values gets most people to the right answer faster and more reliably. The goal isn't to never use algebra — it's to have another path when algebra isn't getting you to the finish line.

Want the full strategy? Read: GMAT® Algebra: What to Do When You're Stuck

From Episode 45 of Real GMAT® Problems (The GMAT® Strategy Podcast).