2012 AMC 10B Problem 5

P1-10~2 minPrint

Ratios, Percents & Averages Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

The tax and the tip are both taken from the meal price, so every dollar of meal costs 1.25 dollars in total. Divide 27.50 by 1.25.

Solution

1. Read the wording carefully: the $15\%$ tip is on the meal price before tax, and the $10\%$ tax is on the meal price before tip. So both are percents of the same number, the meal price.
2. See what one dollar of meal costs in total. It is $1$ dollar for the meal, plus $0.10$ tax, plus $0.15$ tip, which is $1.25$ dollars.
3. So the total bill is $1.25$ times the meal price. Call the meal price $x$ :
$1.25x = 27.50.$
4. Divide to undo the multiplying: $x = 27.50 \div 1.25 = 22$ .
5. Check: tax is $10\%$ of $22$ , which is $2.20$ . Tip is $15\%$ of $22$ , which is $3.30$ . Then $22 + 2.20 + 3.30 = 27.50$ . Correct.

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

Why this works

When several percents are all taken of the same starting amount, you can add them into one multiplier (here $1 + 0.10 + 0.15 = 1.25$ ). To go backward from a total to the starting amount, divide by that multiplier. Subtracting $25\%$ of the total gives the wrong answer, because the $25\%$ was taken of the smaller meal price, not of the total.

Alternative approach

Try the answer choices. Each total is the choice times $1.25$ :

Meal	$18$	$20$	$21$	$22$	$24$
Total	$22.50$	$25.00$	$26.25$	$27.50$	$30.00$

Only $22$ gives $27.50$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(D)}\ 22}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)