"The Shaded Section Comprises What Fractional Part of the Area of the Circular Region?" — GMAT® Worked Solution

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

In the circular region with center $O$ shown above, the two unshaded sections comprise $\frac{3}{7}$ and $\frac{1}{3}$ of the area of the circular region. The shaded section comprises what fractional part of the area of the circular region?

(A) $\dfrac{3}{5}$

(B) $\dfrac{6}{7}$

(C) $\dfrac{2}{21}$

(D) $\dfrac{5}{21}$

(E) $\dfrac{16}{21}$

Try it before reading on.

Step 1: Write What the Question Asks

The diagram shows a circle cut into three sections, two of them labeled with fractions, and the question lives on the third one. So the first line of scratch work is the question itself:

shaded = ?

That line decides the direction of the arithmetic. The two labeled sections are unshaded, which means they get subtracted from the whole circle rather than combined with each other. Everything after this step is fraction mechanics.

Step 2: Set Up the Subtraction From the Whole

A whole circle counts as 1, and 1 can be written as a fraction with any denominator. The setup:

$1 - \frac{3}{7} - \frac{1}{3}$

Subtracting from 1 is the standard move for fractional-part-of-the-whole questions, and the circle is just the packaging.

Step 3: Convert to a Common Denominator

The denominators in play are 7 and 3, and their least common multiple is 21. Each piece gets rewritten over 21:

Each conversion multiplies by a form of 1: $\frac{3}{7} \times \frac{3}{3} = \frac{9}{21}$ and $\frac{1}{3} \times \frac{7}{7} = \frac{7}{21}$. If the common-denominator step has gotten rusty, GMAT® Fractions: A Complete Guide covers the procedure in full.

Step 4: Subtract

$\frac{21}{21} - \frac{9}{21} = \frac{12}{21}$

Then one more subtraction:

$\frac{12}{21} - \frac{7}{21} = \frac{5}{21}$

The answer is (D).

One check before moving on: $21 = 3 \times 7$, and neither factor divides 5, so $\frac{5}{21}$ is already in lowest terms.

Where the Wrong Answers Come From

Choice (E) is $\frac{16}{21}$, and about 3% of test takers pick it. That's the two given fractions added together: $\frac{3}{7} + \frac{1}{3} = \frac{9}{21} + \frac{7}{21} = \frac{16}{21}$. It's the unshaded area, which is the answer to a question the problem didn't ask.

The remaining choices catch fraction-arithmetic slips: a wrong common denominator, an arithmetic error, etc. Choice (E) is the one worth studying, because the error behind it usually isn't arithmetic at all. Under time pressure, the two labeled fractions are the loudest data on the screen, and combining them produces a clean answer to the wrong question.

Why This Problem Matters

Two things make this problem worth a visit.

The first is the trap. Adding $\frac{3}{7}$ and $\frac{1}{3}$ takes work to do and no work to notice: both fractions sit right there in the diagram, and the arithmetic runs cleanly. The habit that catches it costs two seconds. Write what the question asks before any math, and in this case the line shaded = ? points the arithmetic at subtraction from the whole.

The second is what the problem says about geometry on the current exam. This question wears a circle as its costume and solves like a fractions problem, and that's a fair picture of how geometry shows up now: usually one step of geometric reasoning wrapped around other content, rather than the multi-step constructions (a triangle inside a square inside a circle) that older editions featured. Volume and surface area do still come up, so those formulas stay worth knowing. An entire unit on legacy geometry questions is a poorer use of prep time than the same hours spent on the algebra and arithmetic those questions wrap around.

Ready for the next one? "What Is the Value of (0.3)⁵/(0.3)³?" — GMAT® Worked Solution.

Want the full strategy? Read: GMAT® Setup Habits: The Question-First System

From Episode 13 of Real GMAT® Problems (The GMAT® Strategy Podcast).