AMC 8 · 2023 · #13

Grade 4 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

equal-spacingfraction-arithmeticdivisibility-rules systematic-enumerationguess-and-check ↑ Prerequisites: fraction-arithmeticdivisibility-rules

📏 Short solution 💡 2 insights 📊 Diagram

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Problem

A race route runs straight from Start to Finish. Seven water stations and two repair stations are each evenly spaced between Start and Finish. The 3rd water station is 2 miles past the 1st repair station. Find the total race length d in miles.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

We have only five answer choices, and for each candidate length d it is easy to find where the 1st repair station (d ÷ 3 from Start) and the 3rd water station (3 × d ÷ 8 from Start) sit. So we can simply try each choice and keep the one where the gap is exactly 2 miles. A quick diagram first makes the 'divides into 8 (or 3) equal pieces' idea concrete, and the multiple-choice format lets us eliminate the rest instead of solving an algebra equation.

1STEP 1

Draw the route S→F: 7 water ticks split it into 8 equal pieces, 2 repair ticks into 3, so each water gap is $\frac{d}{8}$ , each repair gap $\frac{d}{3}$ .

water gap =

\frac{d}{8}

, repair gap =

\frac{d}{3}

2STEP 2

Position = gap × station number, so the 1st repair station sits at $\frac{d}{3}$ from Start and the 3rd water station at $\frac{3d}{8}$ .

R₁ =

\frac{d}{3}

, W₃ =

\frac{3d}{8}

3STEP 3

Test each choice: find R₁ = d÷3 and W₃ = 3×(d÷8), then check if W₃ - R₁ = 2; the choices 8, 16, 24 give gaps too small.

d=24:& R₁ = 8, W₃ = 9, W₃ - R₁ = 1 ; d=48:& R₁ = 16, W₃ = 18, W₃ - R₁ = 2 ✓ ; d=96:& R₁ = 32, W₃ = 36, W₃ - R₁ = 4

4STEP 4

Only d = 48 gives W₃ - R₁ = 2 miles, matching the condition; the other four choices are ruled out.

d = 48 = (D)

Answer

Check the winner against the original setup. With d = 48, water stations sit at 6, 12, 18, 24, 30, 36, 42 miles (each

\frac{1}{8}

of 48), and repair stations sit at 16 and 32 miles (each

\frac{1}{3}

of 48). The 3rd water station is at 18 and the 1st repair station is at 16, so the water station is exactly 2 miles past the repair station — matches the problem. The race length 48 miles is also a believable distance for a bike race.

💡Key takeaway

This AMC 8 problem only needs Grade 4 division and 'fraction of a whole' you already know!

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