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2012 AMC 10B Problem 5

2012 AMC 10B Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 2 min · position P1-10

Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?

A)
$18$
B)
$20$
C)
$21$
D)
$22$
E)
$24$

Answer

Key insight

Tax and tip are both percentages of the same base price, so the total is 1.25 times the meal: 27.50 / 1.25 = 22.

Solution

Let the meal cost $x$ dollars. The tax is $10\%$ of $x$ and the tip is $15\%$ of $x$ ; both are computed on the same pre-tax, pre-tip price. So the total is
$x + 0.10x + 0.15x = 1.25x = 27.50.$
Dividing, $x = \dfrac{27.50}{1.25} = 22$ .

Check: $22 + 2.20 + 3.30 = 27.50$ .

The answer is $\boxed{\textbf{(D)}\ 22}$ .

Why this works

When several percentages are all taken of the same base, they add into a single multiplier ( $1 + 0.10 + 0.15$ ). The problem's careful wording ("before the sales tax," "on the pre-tip amount") is exactly what guarantees this; read for it. Going from a total back to the base means dividing by the multiplier, never subtracting the percent.

Alternative approach

Test the choices: $25\%$ of $22$ is $5.50$ , and $22 + 5.50 = 27.50$ . The other choices give totals $22.50, 25, 26.25, 30$ .

The trap

Subtracting 25% of the total (27.50 * 0.75 = 20.63) instead of dividing by 1.25, or compounding the tip on top of the tax.

Common mistakes

Subtracting 25% of the total (27.50 * 0.75 = 20.63) instead of dividing by 1.25, or compounding the tip on top of the tax.
Treating the tip as $15\%$ of the taxed amount, giving $1.1 \cdot 1.15 x = 27.50$ and a non-integer $x \approx 21.7$ .

Techniques

Set up the equation/formula and compute; no special trick needed