"One Third of the Cars Sold Were Sedans..." — GMAT® Worked Solution

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

According to a car dealer's sales report, one third of the cars sold during a certain period were sedans and one fifth of the other cars sold were station wagons. If $N$ station wagons were sold during that period, how many sedans, in terms of $N$, were sold?

(A) $\dfrac{2N}{15}$

(B) $\dfrac{3N}{5}$

(C) $\dfrac{5N}{3}$

(D) $\dfrac{5N}{2}$

(E) $\dfrac{15N}{2}$

Try it before reading on.

Step 1: Understand "In Terms Of"

Before doing any math, let's clarify what "in terms of $N$" means, because this phrase causes a lot of confusion and getting it wrong makes the problem much harder than it needs to be.

"In terms of $N$" means: express the answer using only the variable $N$. No other variables allowed. You can see this reflected in the answer choices, which all contain $N$ and numbers but no other variables, even though the problem involves multiple quantities.

To solve, we'll set up equations with all the variables we need, then use substitution to eliminate every variable except $N$.

Step 2: Write What's Given and What's Asked

Let $T$ = total cars sold during the period.

Given:

• One third of $T$ are sedans: $\frac{1}{3}T$
• One fifth of the OTHER cars are station wagons: $\frac{1}{5} \times \frac{2}{3}T$
• $N$ station wagons were sold

Asked:

• Number of sedans, in terms of $N$

Notice the word "other." If one third of the cars are sedans, the other cars are the remaining two thirds. One fifth of those other cars (not one fifth of all cars) are station wagons. Misreading "one fifth of the other cars" as "one fifth of all cars" is the most common error on this problem, and it leads directly to option (C), which is wrong.

Step 3: Set Up the Expressions

Let $S$ = number of sedans:

$S = \frac{1}{3}T$

Let $N$ = number of station wagons. Since station wagons are one fifth of the other cars (which are two thirds of the total):

$N = \frac{1}{5} \times \frac{2}{3}T = \frac{2}{15}T$

Step 4: Eliminate T Through Substitution

We need $S$ in terms of $N$ only, which means $T$ has to disappear. From the station wagon equation, solve for $T$:

$T = \frac{15}{2}N$

Now substitute this expression for $T$ into the sedan equation:

$S = \frac{1}{3} \times \frac{15}{2}N$

Step 5: Solve for S

Multiply the fractions:

$S = \frac{1 \times 15}{3 \times 2}N = \frac{15}{6}N = \frac{5}{2}N$

The answer is (D).

The Wrong Path: What Happens If You Miss "Other"

If you read the problem as "one fifth of all cars are station wagons" instead of "one fifth of the other cars," you'd set up:

$N = \frac{1}{5}T$

Solving for $T$: $T = 5N$

Substituting into the sedan equation: $S = \frac{1}{3} \times 5N = \frac{5}{3}N$

That's option (C), and about 20% of test takers pick it. The math is correct for the wrong reading of the problem. The error isn't in the algebra, it's in the transcription.

This is why double-checking your transcription matters. After writing down what the problem says, look back at the screen and verify each phrase. The word "other" is one word, but it changes the entire setup. A quick reread takes two seconds and catches this error every time.

Why This Problem Matters

About 30% of test takers miss this problem. Two distinct errors account for most of the misses:

First, misreading "one fifth of the other cars" as "one fifth of all cars." This is a transcription error, not a math error. The fix: double-check what you wrote against the screen after transcribing each piece of information.

Second, not understanding what "in terms of $N$" requires. If you don't realize that $T$ needs to disappear, you can do correct math with $T$ in the equation and never match any answer choice. The fix: write the target variable and the allowed variable at the top of your scratchwork, and identify the variable to eliminate before you start solving.

The broader lesson: on word translation problems, the setup is everything. The algebra is usually straightforward once the equations are correct. Most points are lost in the translation from English to math, not in the computation that follows. Building a habit of careful transcription and double-checking saves more points than learning any new formula.

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).