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, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$ . To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Maximum heart rate: $M = 220 - \text{age}$; Target heart rate: $T = 80\% \text{ of } M$; Athlete's age: $26$ years; Answer choices: (A) $134$, (B) $155$, (C) $176$, (D) $194$, (E) $243$

Unknowns: The target heart rate $T$, rounded to the nearest whole number

Understand

Givens: Maximum heart rate: $M = 220 - \text{age}$; Target heart rate: $T = 80\% \text{ of } M$; Athlete's age: $26$ years; Answer choices: (A) $134$, (B) $155$, (C) $176$, (D) $194$, (E) $243$

Plan

Primary tool: #3 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.

Execute — Answer: B

#3 Set Up an Equation 6.EE.A.2 Step 1

Translate the first sentence into an expression for the maximum heart rate $M$.
Subtract the athlete's age, $26$, from $220$.

$$M = 220 - 26 = 194$$

💡 Naming the maximum with a letter makes the second step easy to write: "$80\%$ of $M$."

#3 Set Up an Equation 6.RP.A.3 Step 2

Translate the second sentence into an expression for the target heart rate $T$.
Take $80\%$ of $M$ by multiplying by $0.80$ (or equivalently $\tfrac{4}{5}$).

$$T = 0.80 \times 194 = 155.2$$

💡 "$80\%$ of" is the Grade 6 percent move: convert $80\%$ to $0.80$, then multiply.

#3 Set Up an Equation 5.NBT.A.4 Step 3

Round to the nearest whole number.
The digit in the tenths place is $2$, which is less than $5$, so round down.

$$T \approx 155 \;\Rightarrow\; \textbf{(B)}$$

💡 Standard rounding rule: tenths digit under $5$ means keep the whole-number part as-is.

[1] #3 6.EE.A.2 Translate the first sentence into an expression for the maximum heart rate $M$.

[2] #3 6.RP.A.3 Translate the second sentence into an expression for the target heart rate $T$.

[3] #3 5.NBT.A.4 Round to the nearest whole number. The digit in the tenths place is $2$, which i

Review

Reasonableness: 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 $150$s. $155$ 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.

Alternative: Tool #9 (Solve an Easier Problem): use friendlier numbers to see the size. Round $194$ to $200$ and take $80\%$: $0.80 \times 200 = 160$. The exact answer should be a little under $160$ since we rounded $194$ up to $200$. Among the choices, only (B) $155$ fits "a little under $160$," confirming the answer without finishing the exact arithmetic.

CCSS standards used (min grade 6)

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Naming the maximum heart rate $M = 220 - 26$ and the target $T = 0.80 \times M$ so the two-step recipe becomes two clean expressions.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world problems, including finding a percent of a quantity (Computing "$80\%$ of $194$" as $0.80 \times 194 = 155.2$.)
5.NBT.A.4 Use place value understanding to round decimals to any place (Rounding $155.2$ to the nearest whole number to get $155$.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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