AMC 10 · 2012 · #5

Grade 7 arithmetic
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery
percentagelinear-equations-one-var convert-to-algebra ↑ Prerequisites: percentage
📏 Medium solution 💡 2 insights
Problem
Anna eats dinner at a restaurant where the sales tax on meals is 10% of the price of the meal. She leaves a tip of 15% of that same price before tax, and the tax is calculated on the pre-tip amount, so both charges are figured from the same base price. She spends a total of $27.50 for dinner. Find the cost of her dinner without tax or tip, in dollars.

Pick an answer.

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

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The base price is the one unknown that everything else is built from, so Tool #4 (Introduce a Variable) names it once and writes the tip and tax as 0.15x and 0.10x. Tool #13 (Convert to Algebra) then turns the sentence “meal plus tip plus tax equals 27.50” into a single equation and collects the like terms into one multiple of x. Tool #3 (Eliminate Possibilities) checks the result against the built-in trap answers, such as the value you get if you wrongly tax the tip too.

1STEP 1

Name the base price

Let x be the dinner's price before tax and tip; then the tip is 0.15x and the tax is 0.10x.

x=base price, tip=0.15x, tax=0.10x
2STEP 2

Add the parts into one equation

Meal plus tip plus tax is x+0.15x+0.10x, and collecting like terms gives 1.25x=27.50.

x+0.15x+0.10x=1.25x=27.50
3STEP 3

Solve for the base price

Divide both sides by 1.25, which is the same as multiplying by 45\frac{4}{5}, giving x=22.

x=27.50/1.25=22
4STEP 4

Check the total and pick the answer

Check: the tip is 3.30, the tax is 2.20, and 22+3.30+2.20=27.50 matches, so the answer is (D).

22+3.30+2.20=27.50 → (D)
Answer
22
The base price must be well under the $27.50 total, since a quarter of it is added on as tip and tax. A quick estimate: 25% extra on about $22 is roughly $5.50, and 22+5.50=27.50 lands exactly on the total, so $22 is consistent. Choice (E) 24 would already push the total past 27.50 once the 25% is added, and the smaller choices leave the total short, so only (D) fits.
💡Key takeaway

When a tip and a tax are both taken from the same price, add their percents to the whole and divide the total by that single factor.

  • Name the base price
  • Add the parts into one equation
  • Solve for the base price
  • Check the total and pick the answer

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