AMC 8 · 2004 · #7

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

percentageformula-substitutionmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmeticfraction-decimal-conversion

📏 Short solution 💡 1 insight

Problem

An athlete's target heart rate equals 80% of the maximum heart rate, and the maximum heart rate is 220 minus the athlete's age. For a 26-year-old, what is the target heart rate, rounded to the nearest whole number?

Pick an answer.

(A)

134

(B)

155

(C)

176

(D)

194

(E)

243

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Set Up an Equation

The problem hands us a two-step recipe in words: subtract age from 220, then take 80%. Tool #3 (Set Up an Equation) is the cleanest match — translate each English sentence into an arithmetic expression and evaluate in order. No invariant, no pattern, no guessing required. Just two operations and a rounding step.

1STEP 1

Subtract the age from 220: the maximum heart rate M is 194.

M = 220 - 26 = 194

2STEP 2

The target is 80% of M: multiply 194 by 0.80 to get 155.2.

T = 0.80 × 194 = 155.2

3STEP 3

The tenths digit 2 is less than 5, so round 155.2 down to 155.

T ≈ 155 → (B)

Answer

155

Sanity check the size. The maximum heart rate 194 should land between (D) 194 (the maximum itself, a trap) and (A) 134. Eighty percent of 194 is a bit less than 194 but well above half of it (97), so the answer should sit in the upper 150s. 155 fits. The trap answers map to predictable mistakes: (D) 194 skips the 80% step, (A) 134 comes from 194 - 60 (subtracting age twice or misreading 80% as 40%), (C) 176 would be 80% of 220 (forgetting to subtract age), and (E) 243 is roughly 80% added on top of 194 instead of taken from it.

💡Key takeaway

Word-recipe problems become easy once you write each sentence as an expression. Here: 220 - 26 = 194, then 0.80 × 194 = 155.2, then round to 155. Three small steps, no algebra needed.

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