"A Store Currently Charges the Same Price for Each Towel That It Sells..." — GMAT® Worked Solution

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

A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel?

(A) $1

(B) $2

(C) $3

(D) $4

(E) $12

Try it before reading on.

Step 1: Write What's Given and Asked

Given:

• The current price per towel is unknown
• If the price increases by $1, you can buy 10 fewer towels for $120
• The total budget is $120 in both scenarios

Asked: the current price of each towel

Step 2: Recognize the Comparison and Infer the Implicit Original

The problem describes a "new" situation (price increased, 10 fewer towels), which means there's also a "current" situation that isn't explicitly described. This is the key inference: if there's a new price and new quantity for $120, there must be a current price and current quantity for the same $120.

Set up a table with "Current" and "New" columns:

The word "current" in the problem is the cue. If the problem mentions a "current price" and then describes what happens if that price changes, the current situation is implied even though the problem doesn't give you numbers for it directly.

Step 3: Write the Equations From the Table

Each column gives you an equation. Price times quantity equals total spent.

From the Current column:

$pt = 120$

From the New column:

$(p + 1)(t - 10) = 120$

Step 4: Solve by Substitution

From the first equation, isolate $t$:

$t = \frac{120}{p}$

Substitute into the second equation:

$(p + 1)\left(\frac{120}{p} - 10\right) = 120$

Distribute the left side:

$\frac{120(p + 1)}{p} - 10(p + 1) = 120$

$120 + \frac{120}{p} - 10p - 10 = 120$

Subtract 120 from both sides:

$\frac{120}{p} - 10p - 10 = 0$

Multiply through by $p$ to clear the fraction:

$120 - 10p^2 - 10p = 0$

Divide by $-10$:

$p^2 + p - 12 = 0$

Step 5: Factor the Quadratic

Factor the quadratic equation:

$(p + 4)(p - 3) = 0$

This gives two solutions: $p = -4$ or $p = 3$.

Since a price can't be negative, $p = 3$.

The answer is (C).

Why This Problem Matters

About 25% of test takers miss this problem, and the most popular wrong answer is (D) $4.

Here's why: after correctly factoring to $(p + 4)(p - 3) = 0$, the two roots are $p = -4$ and $p = 3$. The common error is solving $p + 4 = 0$ as $p = 4$ instead of $p = -4$, forgetting to move the constant to the other side and negate it. If you make that sign error, you get $4 instead of $3 as your positive root, and $4 is sitting right there in the answer choices.

A price can't be negative, so $p = -4$ is eliminated. That leaves $p = 3$, which is answer choice (C). But if you're moving fast and make the sign error on the first factor, $4 looks like a reasonable towel price and it matches answer choice (D).

The other common struggle on this problem isn't a wrong answer, it's getting stuck at the beginning. The problem never explicitly describes the current situation. It only tells you about the new situation. You have to infer that the current price and current quantity also relate to $120. Without that inference, you only have one equation with two variables and can't solve.

The "Current vs. New" table makes the implicit original visible. When you see an empty "Current" column, you fill it in. When both columns are complete, both equations emerge naturally. The table does the inference work that the problem is testing.

Want the full strategy behind this problem? Read: GMAT® Word Translations: Three Systems for Setup and Organization

From Episode 34 of Real GMAT® Problems (The GMAT® Strategy Podcast).