AMC 10 · 2017 · #14

Grade 8 rate-ratio
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery
percentagesystems-of-equationslinear-equations-two-var convert-to-algebra ↑ Prerequisites: linear-equations-two-var
📏 Medium solution 💡 2 insights
Problem
Roger's allowance is A dollars. He buys a movie ticket and a soda. The ticket costs 20% of the difference between A and the soda's price; the soda costs 5% of the difference between A and the ticket's price. Rounded to the nearest whole percent, what percent of A is spent on the ticket and soda together?

Pick an answer.

(A)
$9\%$
(B)
$19\%$
(C)
$22\%$
(D)
$23\%$
(E)
$25\%$

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Each price is described in terms of the other price, so naming both prices is the unlock. Tool #4 (Introduce a Variable) lets m stand for the movie ticket cost and s for the soda cost. Tool #13 (Convert to Algebra) turns the two English sentences into two equations — each percent-of-a-difference becomes a clean linear equation — and because the two equations share the same two unknowns, substituting one into the other (a system of two linear equations) pins down both. Tool #8 (Analyze the Units) keeps the goal in view: we want (m+s)/A as a percent, so A must cancel and the dollars turn into a pure ratio.

1STEP 1

Name the two costs

Let m be the ticket cost and s the soda cost; each sentence becomes an equation (20% = 0.2, 5% = 0.05).

m = 0.2 (A - s), s = 0.05 (A - m)
2STEP 2

Expand the percents

Distribute each decimal across the difference: 20% of (A-s) becomes 0.2A - 0.2s, and 5% of (A-m) becomes 0.05A - 0.05m.

m = 0.2A - 0.2s, s = 0.05A - 0.05m
3STEP 3

Substitute to isolate the ticket

Substitute s = 0.05A - 0.05m into the first equation and collect terms: 0.99m = 0.19A, so m = 19/99A.

m = 0.19A + 0.01m → 0.99m = 0.19A → m = 19/99A
4STEP 4

Back-solve for the soda

Plug m = 19/99A back into s = 0.05A - 0.05m to get s = 4/99A.

s = 0.05A - 0.05·19/99A = (4.95 - 0.95)/99A = 4/99A
5STEP 5

Add and convert to a percent

Add m + s = 19/99A + 4/99A = 23/99A; A cancels, leaving 23/99 ≈ 23%, choice (D).

(m+s)/A = 23/99 = 0.2323… ≈ 23% → (D)
Answer
23%
The two pieces are 19/99A ≈ 19.2% and 4/99A ≈ 4.0%, and 19.2% + 4.0% = 23.2%, matching 23/99. A sanity check on the definitions: the ticket should be the bigger spend (it is 20% of almost all of A), and the soda small (5% of almost all of A) — exactly what 19% versus 4% shows. Rounding 23.23% gives 23%, choice (D), and the nearby trap 19% (the ticket alone) is correctly excluded because the question asks for both items.
💡Key takeaway

When two unknown prices each depend on the other, give them letters, turn each sentence into an equation, and substitute one into the other so the allowance cancels out.

  • Name the two costs
  • Expand the percents
  • Substitute to isolate the ticket
  • Back-solve for the soda
  • Add and convert to a percent

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