"Last Sunday a Certain Store Sold Copies of Newspaper A..." — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 13th Edition

Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If $r$ percent of the store's revenues from newspaper sales was from Newspaper A and if $p$ percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses $r$ in terms of $p$?

(A) $\dfrac{100p}{125 - p}$

(B) $\dfrac{150p}{250 - p}$

(C) $\dfrac{300p}{375 - p}$

(D) $\dfrac{400p}{500 - p}$

(E) $\dfrac{500p}{625 - p}$

Try it before reading on.

Why This Problem Is Hard

This problem has two things that make people freeze:

Variables in the answer choices. The answers aren't numbers — they're formulas with $p$. That means you can't just solve and match. You have to either express $r$ in terms of $p$ algebraically, or pick numbers and plug in.

Two related but different quantities. $p$ is about copies sold. $r$ is about revenue. Same newspapers, different units. Mixing them up is where most errors happen.

The good news: there are two clean approaches. We'll show both.

Approach 1: Pick Numbers

When you see variables in the answer choices, picking numbers is almost always faster than algebra. The key is picking numbers that make the arithmetic easy.

Step 1: Pick Simple Values

Let's say the store sold 4 copies of Newspaper A and 4 copies of Newspaper B.

Why 4? Because Newspaper B costs $1.25, and $1.25 \times 4 = 5.00$ — a clean number. We could pick any positive integers, but 4 makes the math work nicely.

Step 2: Calculate $p$

$p$ is the percent of newspapers sold that were copies of Newspaper A.

Total newspapers sold: $4 + 4 = 8$

Newspaper A copies: $4$

$p = \frac{4}{8} \times 100 = \mathbf{50}$

Step 3: Calculate $r$

$r$ is the percent of revenue that came from Newspaper A.

Revenue from A: $4 \times 1.00 = 4.00$

Revenue from B: $4 \times 1.25 = 5.00$

Total revenue: $4.00 + 5.00 = 9.00$

$r = \frac{4}{9} \times 100 = \mathbf{\frac{400}{9}}$

Step 4: Plug $p = 50$ Into the Answer Choices

The correct answer should give us $\frac{400}{9}$ when we plug in $p = 50$.

(A) $\frac{100(50)}{125 - 50} = \frac{5000}{75} = 66.67$ — not $\frac{400}{9}$

(B) $\frac{150(50)}{250 - 50} = \frac{7500}{200} = 37.5$ — not $\frac{400}{9}$

(C) $\frac{300(50)}{375 - 50} = \frac{15000}{325} \approx 46.15$ — not $\frac{400}{9}$

(D) $\frac{400(50)}{500 - 50} = \frac{20000}{450} = \frac{2000}{45} = \mathbf{\frac{400}{9}}$ ✓

(E) $\frac{500(50)}{625 - 50} = \frac{25000}{575} \approx 43.48$ — not $\frac{400}{9}$

The answer is (D).

Shortcut: Divisibility Check

Before plugging into all five choices, look at the denominators. We know the correct answer simplifies to $\frac{400}{9}$, so the denominator (before simplifying) must be divisible by 9.

• (A) $125 - 50 = 75$. Is 75 divisible by 9? No.
• (B) $250 - 50 = 200$. Is 200 divisible by 9? No.
• (C) $375 - 50 = 325$. Is 325 divisible by 9? No.
• (D) $500 - 50 = 450$. Is 450 divisible by 9? Yes. ✓
• (E) $625 - 50 = 575$. Is 575 divisible by 9? No.

Only (D) passes. If you spot this, you skip four calculations entirely.

Approach 2: Algebra

For those who prefer algebra — or if the numbers don't work out as cleanly — here's the setup.

Step 1: Define Variables

Let $a$ = number of copies of Newspaper A sold

Let $b$ = number of copies of Newspaper B sold

Step 2: Express $p$

$p = \frac{a}{a + b} \times 100$

Step 3: Express $r$

Revenue from A: $a \times 1.00 = a$

Revenue from B: $b \times 1.25 = \frac{5b}{4}$

Total revenue: $a + \frac{5b}{4}$

$r = \frac{a}{a + \frac{5b}{4}} \times 100$

Simplify the denominator:

$a + \frac{5b}{4} = \frac{4a + 5b}{4}$

So:

$r = \frac{a}{\frac{4a + 5b}{4}} \times 100 = \frac{4a}{4a + 5b} \times 100 = \mathbf{\frac{400a}{4a + 5b}}$

Step 4: Express $r$ in Terms of $p$

From $p = \frac{100a}{a + b}$, we need to eliminate $a$ and $b$ and get everything in terms of $p$.

From $p$: $a + b = \frac{100a}{p}$, so $b = \frac{100a}{p} - a = a\left(\frac{100}{p} - 1\right) = \frac{a(100 - p)}{p}$

Substitute into $r$:

$r = \frac{400a}{4a + 5b}$

$= \frac{400a}{4a + 5 \times \frac{a(100 - p)}{p}}$

Factor out $a$:

$= \frac{400a}{a\left(4 + \frac{5(100 - p)}{p}\right)}$

The $a$ cancels:

$= \frac{400}{4 + \frac{5(100 - p)}{p}}$

Simplify the denominator:

$4 + \frac{5(100 - p)}{p} = 4 + \frac{500 - 5p}{p} = \frac{4p + 500 - 5p}{p} = \frac{500 - p}{p}$

So:

$r = \frac{400}{\frac{500 - p}{p}} = \mathbf{\frac{400p}{500 - p}}$

The answer is (D).

Why This Problem Matters

This problem tests two skills that show up across the GMAT® quant section:

Picking numbers when variables are in the answer choices. This is one of the highest-value techniques on the entire exam. When you see formulas with variables in the answers, picking numbers turns abstract algebra into arithmetic. The trick is picking numbers that make the math clean — like choosing 4 copies here because $1.25 \times 4 = 5$.

Keeping track of what each variable represents. $p$ is about units sold. $r$ is about revenue. They use the same underlying data (newspapers A and B) but measure different things. Most wrong answers come from mixing up which is which — calculating revenue percent when you meant units, or vice versa.

The divisibility shortcut is worth remembering. When you pick numbers and the result is a fraction (like $\frac{400}{9}$), you can eliminate answer choices whose denominators aren't divisible by 9 before doing any real arithmetic. On a timed exam, saving 60 seconds on one problem matters.

Want the full strategy behind picking numbers on GMAT® word problems? Read: GMAT® Word Problems: When Picking Numbers Beats Algebra