"There Are Four More Women Than Men on Centerville's Board of Education..." — GMAT® Worked Solution

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

There are four more women than men on Centerville's board of education. If there are 10 members on the board, how many are women?

(A) 3

(B) 4

(C) 6

(D) 7

(E) 8

Try it before reading on.

Step 1: Translate the Relationships Into Equations

The problem gives us two relationships, and our job is to turn each one into an equation.

Let $m$ = the number of men and $w$ = the number of women.

Relationship 1: "There are four more women than men."

$w = m + 4$

Relationship 2: "There are 10 members on the board."

$m + w = 10$

That second one might feel less obvious because it's not stated as an equation. But "10 members total" means the number of men plus the number of women equals 10. Train yourself to recognize this kind of hidden equation in word problems. A statement about a total quantity is almost always an equation waiting to be written.

Step 2: Isolate the Variable You're NOT Solving For

The question asks for the number of women, so we need to solve for $w$. That means we should isolate $m$ first, so it gets eliminated when we substitute.

Take the first equation and solve for $m$:

$w = m + 4$

$m = w - 4$

Step 3: Substitute and Solve

Now replace $m$ in the second equation with $w - 4$:

$w + (w - 4) = 10$

$2w - 4 = 10$

$2w = 14$

$w = 7$

The answer is (D).

There are 7 women on the board. You can verify: if there are 7 women, there are $10 - 7 = 3$ men, and $7 - 3 = 4$, which matches "four more women than men."

Step 4: The Guess-and-Test Alternative

If setting up the algebra doesn't click, try plugging in the answer choices. Pick an answer, treat it as the number of women, and check whether it satisfies both conditions.

Test (D): If there are 7 women, then there are $10 - 7 = 3$ men. Are there four more women than men? $7 - 3 = 4$. Yes. (D) works.

This strategy is especially useful when the answer choices are simple integers and the algebra is complex or you're unsure of your setup. It can be a real lifeline when the textbook approach isn't clicking. It's not a silver bullet for every problem, but for word problems with relatively simple answer choices, it often works well.

Why This Problem Matters

About 10% of test takers miss this problem, and the most popular wrong answer is (A): 3.

That's the number of men on the board. If you do all the algebra correctly but solve for $m$ instead of $w$, you get 3 instead of 7. The math is right. The variable is wrong.

This is one of the most common and most preventable errors in GMAT® quant. The fix is a habit, not a formula: write what's asked at the top of your scratchwork before doing any math. "How many are women?" means $w$ goes at the top of your work. Then isolate the other variable first, so the one you don't want disappears during substitution.

The guess-and-test approach also prevents this error, because you're testing for the specific quantity the question asks about. You can't accidentally solve for men when you're plugging in candidate values for women.

Want the full strategy behind this problem? Read: GMAT® Quant: Three Systems for Algebra, Compound Interest, and Estimation

From Episode 22 of Real GMAT® Problems (The GMAT® Strategy Podcast).

Ready for the next one? "Leona Bought a 1-Year $10,000 Certificate of Deposit..." — GMAT® Worked Solution.