"If m > 0 and x is m% of y, Then in Terms of m, y is What Percent of x?" — GMAT® Worked Solution
From Episode 39 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind percent word problems on the GMAT®, read: Translating Percent Word Problems on the GMAT®: A One-Word-at-a-Time System.
The Problem
Source: Official Guide for GMAT® Review, 11th Edition
If m > 0 and x is m% of y, then in terms of m, y is what percent of x?
(A) 100m
(B) 1⁄100m
(C) 1⁄m
(D) 10⁄m
(E) 10,000⁄m
Translate the First Statement One Word at a Time
The problem gives one fact and asks for one quantity. Translate each separately.
First fact: "x is m% of y"
- "x" → x
- "is" → =
- "m" → m
- "percent" → ÷ 100
- "of" → ×
- "y" → y
That gives us:
x = m⁄100 × y
Note the m sits over 100 — not on its own. This is the most important step in the whole problem. We will come back to why in the "Why This Problem Matters" section.
Translate the Question One Word at a Time
The question: "y is what percent of x"
- "y" → y
- "is" → =
- "what" → a new variable. Use P
- "percent" → ÷ 100
- "of" → ×
- "x" → x
That gives us:
y = P⁄100 × x
Now we have two equations:
- x = m⁄100 × y
- y = P⁄100 × x
We want to solve for P in terms of m. The phrase "in terms of m" means P should be expressed using only m — no x, no y. The answer choices confirm this: they have only m in them.
Step 1: Substitute to Eliminate x
Take equation 2:
y = P⁄100 × x
Replace x with what equation 1 says x equals: m⁄100 × y.
y = P⁄100 × (m⁄100 × y)
x is gone. Now we have one equation with y, m, and P.
Step 2: Eliminate y
Divide both sides by y. The problem implies y ≠ 0 (since x is some percent of y), so this is safe:
1 = P⁄100 × m⁄100
y is gone. We are left with m and P.
Step 3: Clear the Fractions
Multiply both sides by 100 × 100 = 10,000:
10,000 = P × m
Step 4: Solve for P
Divide both sides by m:
P = 10,000⁄m
The answer is (E).
Why This Problem Matters
About 31% of test takers miss this one — roughly three times the miss rate of Problem 1. The math is not much harder. What changes is one specific translation step that catches a lot of test takers off guard.
The biggest trap: treating m% as a single quantity instead of m ÷ 100.
Two versions of the mistake show up:
The plugging-in version. Test taker decides to plug in for m to make the problem concrete. Wants to test 20%. So sets m = 0.2. But the problem says "m%" — meaning m is the percent itself. If you want 20%, m should equal 20, not 0.2. Setting m = 0.2 means you are actually testing 0.2%, which is a different problem.
The algebra version. Test taker translates "x is m% of y" as x = m × y instead of x = m/100 × y. This skips the divide-by-100 step. From there, the algebra leads to (C), 1⁄m, which picks up about 15% of test takers.
The four-word translation system catches both versions of the mistake. The word percent has its own translation: ÷ 100. It is not part of the variable. Keeping it as a separate step means m stays m, and the divide-by-100 happens on its own.
A few more things worth noting:
The phrase "in terms of m." This is GMAT® shorthand for "express the answer using only m" — no other variables. The answer choices reflect this. Whenever you see "in terms of [variable]," check the answer choices to confirm what variables should and should not appear.
Substitution as a tool for making variables disappear. The problem has three variables (x, y, m) and the answer needs to have one (m). Substitution gets rid of the others. We used equation 1 to replace x in equation 2, then divided by y to eliminate y. Two substitutions, two variables removed.
Variables in the answer choices are a signal. When the answers contain m rather than numbers, the final answer probably will too. Do not be thrown off by ending up with a fraction over m.
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