AMC 10 · 2006 · #7

Grade 7 algebra
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Chained Ratio Linking
percentagelinear-equations-one-varfloor-function convert-to-algebra ↑ Prerequisites: percentage
📏 Medium solution 💡 2 insights
Problem
Three ages are chained by two percent comparisons and add to 23.2 years. Find the number the first person turns on her next birthday.

Pick an answer.

(A)
7
(B)
8
(C)
9
(D)
10
(E)
11
How to solve
Strategy Convert to Algebra

Two sentences of English hide two multiplications, so Tool #13 (Convert to Algebra) leads: rewrite each comparison as a multiplier. Tool #15 (Organize Information in More Ways) decides which age to build on — anchoring on Danielle turns the chain into one variable with no division. Tool #4 (Introduce a Variable) then reduces everything to a single equation. Tool #8 (Analyze the Units) closes the problem: the number that falls out is a length of time in years, and the question asks about birthdays, not about that number. That last translation is where the problem is actually won or lost, so it gets its own step.

1STEP 1

Make each comparison a multiplier

Each comparison becomes a single multiplier.

m = 1.2s s = 0.6d
2STEP 2

Anchor everything on Danielle

Anchoring on the end of the chain writes all three ages with one letter.

m = 1.2(0.6d) = 0.72d
3STEP 3

Solve, then check the fit

The total gives one linear equation with exactly one root.

2.32d = 23.2 → d = 10, s = 6, m = 7.2
4STEP 4

Read 7.2 as an age

A fractional age means the next birthday number is 8, choice (B).

⌊ 7.2 ⌋ + 1 = 7 + 1 = 8 → (B)
Answer
8
The three ages come out as Danielle 10, Sally 6, Mary 7.2, and they hold together: Mary sits above Sally, which matches being older than her, and the ugly total 23.2 was a signal that a clean d = 10 was waiting behind 2.32d. Substituting back reproduces all three given facts exactly, so the values are sufficient, not merely forced. For the final answer, 7 ＜ 7.2 ＜ 8 knocks out (C), (D) and (E) at once, and (A) 7 is the number Mary already turned rather than the one coming, leaving 8.
💡Key takeaway

Turn each percent sentence into a multiplier, build everything on the one person nobody is compared to, and remember your next birthday is the first whole year your age has not reached.

  • Make each comparison a multiplier
  • Anchor everything on Danielle
  • Solve, then check the fit
  • Read 7.2 as an age

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