"The Price of Lunch for 15 People Was $207..." — GMAT® Worked Solution

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

The price of lunch for 15 people was $207 including a 15% gratuity for service. What was the average price per person excluding the gratuity?

(A) $11.73

(B) $12.00

(C) $13.80

(D) $14.00

(E) $15.87

Try it before reading on.

Step 1: Understand the Setup

The total of $207 includes a 15% gratuity added to an unknown original price. We need to find that original price, then divide by 15 to get the per-person average excluding the gratuity.

The critical question: how do we remove the gratuity correctly?

Step 2: Avoid the Trap

The instinct is to take 85% of $207. If the total includes a 15% markup, just subtract 15%, right?

That logic is wrong, and it's the trap that catches about 21% of test takers on this problem.

Here's why it fails. Increasing a number by 15% isn't the same as decreasing the result by 15%. The increased number is larger, so 15% of that larger number is a bigger amount than 15% of the original.

Think about a simpler example. A $100 item with 10% sales tax costs $110. Taking 10% off $110 gives $99, not $100. The same percentage of a larger number produces a larger result, so subtracting it "undoes" too much.

The correct approach: set up an equation where an unknown original price is increased by 15%, and the result equals $207.

Step 3: Set Up the Equation

Let $P$ equal the original total price before gratuity. Increasing $P$ by 15% means:

$\frac{115}{100} \times P = 207$

Alternatively, using the percent change formula $\frac{\text{new} - \text{old}}{\text{old}} \times 100 = \text{percent change}$:

$\frac{207 - P}{P} \times 100 = 15$

Both equations are equivalent. The first is more direct for this problem type.

Step 4: Solve for P

Using the first equation:

$P = 207 \times \frac{100}{115}$

Simplify the fraction $\frac{100}{115}$. Both numbers are divisible by 5:

$\frac{100}{115} = \frac{20}{23}$

Now we have:

$P = 207 \times \frac{20}{23}$

Is 207 divisible by 23? On the GMAT® (no calculator), it's a safe bet that unfamiliar numbers will factor cleanly. Let's try the long division:

    9
   ----
23 ) 207
    207
    ---
      0

$23 \times 9 = 207$. So $207 \div 23 = 9$.

Now substitute back:

$P = 9 \times 20 = 180$

The original price for 15 people was $180.

Step 5: Find the Per-Person Average

Divide the original total by 15:

$\frac{180}{15} = 12$

The average price per person excluding the gratuity is $12.

The answer is (B).

Step 6: Prime Factoring as a Computation Tool

If the fraction arithmetic felt uncertain, prime factoring makes the cancellation visible. Here's how it works on the key numbers:

Prime factor 100:

$100 = 2 \times 50 = 2 \times 2 \times 25 = 2 \times 2 \times 5 \times 5 = 2^2 \times 5^2$

Prime factor 115:

$115 = 5 \times 23$

Prime factor 207:

$207 = 3 \times 69 = 3 \times 3 \times 23 = 3^2 \times 23$

Now rewrite the original computation:

$P = \frac{207 \times 100}{115} = \frac{(3^2 \times 23) \times (2^2 \times 5^2)}{5 \times 23}$

Cancel the 23 and one 5:

$P = \frac{3^2 \times 2^2 \times 5}{1} = 9 \times 4 \times 5 = 180$

The prime factors make the cancellation explicit. No guesswork, no trial division. This technique is valuable whenever you're dividing unfamiliar numbers on the GMAT® and you're not sure what's divisible by what.

Why This Problem Matters

About 30% of test takers miss this problem, and 21% go for option (A), $11.73. That's the answer you get from taking 85% of $207 and dividing by 15. The computation is executed correctly, but the setup is wrong from the start.

What makes this painful is that the wrong setup requires MORE computation than the right one. The people who go down the wrong path do harder math and still get the wrong answer. The correct setup ($\frac{115}{100} \times P = 207$) leads to cleaner numbers and less arithmetic.

The broader lesson: setup determines outcome. A wrong setup with perfect computation still produces a wrong answer. And on the GMAT®, wrong setups are designed to match wrong answer choices, so you won't catch the error by checking whether your answer "looks right."

The fix is a habit: when a problem involves a percent increase or decrease, identify which value is the original (old) and which is the result (new) before writing any equation. If the original is unknown, it goes on one side of the equals sign by itself. The known total goes on the other side.

If you find yourself naturally subtracting the percentage from the total, build flashcards on percent setup. Redo this problem until the correct setup feels automatic.

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).

Ready for the next one? "One Third of the Cars Sold Were Sedans..." — GMAT® Worked Solution.