AMC 8 · 2001 · #5

Grade 6 rate-ratioarithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateunit-conversionestimationmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: rateunit-conversion

📏 Medium solution 💡 3 insights

Problem

Snoopy sees a lightning flash and hears the thunder 10 seconds later. Sound travels at 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far the lightning was from Snoopy.

Pick an answer.

(A)

(B)

$1\frac{1}{2}$

(C)

(D)

$2\frac{1}{2}$

(E)

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

The numbers come dressed in three different units — seconds, feet per second, feet per mile — but the answer must be in miles. Tool #8 (Analyze the Units) gives the recipe: multiply feet/second by seconds to cancel "second" and land on feet, then divide by feet/mile to cancel "feet" and land on miles. Tool #7 (Identify Subproblems) splits that recipe into three clean steps — (a) feet of sound travel, (b) feet → miles, (c) round to the nearest half-mile — so each line does one job.

1STEP 1

Multiply speed by time — feet/second × seconds cancels "second" and leaves feet: 1088 × 10 = 10,880 ft.

1088 ft/s × 10 s = 10,880 ft

2STEP 2

Divide feet by 5280 feet/mile — "feet" cancels and leaves miles: 10,880 ÷ 5280 ≈ 2.06 mi.

(10,880 ft)/(5280 ft/mile) =

\frac{10,880}{5280}

mi ≈ 2.06 mi

3STEP 3

Nearby half-mile marks are 2 and 2 $\frac{1}{2}$ ; 2.06 sits only 0.06 from 2, so round to 2 miles → (C).

|2.06 - 2| = 0.06, |2.06 - 2.5| = 0.44 → 2 mi is closer → (C)

Answer

Quick mental check using the well-known rule of thumb "sound travels about 1 mile every 5 seconds." In 10 seconds, the sound covers roughly

\frac{10}{5}

= 2 miles — exactly the answer (C). The exact speed 1088 ft/s gives 1088 × 5 = 5440 ft per 5 seconds, which is just 160 ft over a mile, so the rule of thumb slightly underestimates and our 2.06 miles is the small correction. Either way, 2 is the nearest half-mile.

💡Key takeaway

Let the units guide you: seconds times feet-per-second gives feet, then dividing by feet-per-mile gives miles. Sound covers 10,880 ft in 10 s, which is about 2.06 mi — closest to 2 miles, answer (C).

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