AMC 8 · 2018 · #17

Grade 5 rate-ratio

Archetype Meeting Time with Piecewise Rates └ Sub-archetype Constant-Rate Closing Speed

rateratio-proportionunit-conversion dimensional-analysisidentify-subproblems ↑ Prerequisites: ratio-proportionunit-conversion

📏 Medium solution 💡 3 insights

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Problem

Bella walks from her house toward Ella's house while Ella bikes toward Bella's house at the same moment. Ella's biking speed is 5 times Bella's walking speed, and Bella covers 2

\frac{1}{2}

feet per step. Their houses are 2 miles (10,560 feet) apart. How many steps has Bella taken by the time they meet?

Pick an answer.

(A)

704

(B)

845

(C)

1056

(D)

1760

(E)

3520

AMC 8 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

We are never told the actual speed in mph or how long the trip takes, so chasing time as a variable is harder than it needs to be. Tool #9 (Easier Related Problem) lets us replace the awkward unit of "time" with a much friendlier unit: "one Bella-step." In the duration of one Bella-step, Bella moves 2.5 ft and Ella — being 5 times faster — moves 5 × 2.5 = 12.5 ft, so the gap between them shrinks by a fixed amount per Bella-step. Tool #8 (Analyze the Units) then keeps the bookkeeping clean: dividing total feet by feet-per-step cancels to leave plain "steps," exactly what the question asks for.

1STEP 1

Swap clock-time for an easier unit — one "Bella-step" — during which Bella moves her step length, 2.5 ft.

Bella per step = 2.5 ft

2STEP 2

In that same Bella-step, Ella rides five times as far — 5 × 2.5 = 12.5 ft, a "times as fast" comparison.

Ella per Bella-step = 5 × 2.5 = 12.5 ft

3STEP 3

Moving toward each other, the gap shrinks each Bella-step by both distances added: 2.5 + 12.5 = 15 ft.

Gap closed per Bella-step = 2.5 + 12.5 = 15 ft

4STEP 4

Divide total feet by feet-per-step; "ft" cancels, leaving steps: $\frac{10,560}{15}$ = 704 steps → (A).

\frac{10,560 ft}{15 \frac{ft}{step}}

= 704 steps → (A)

Answer

704

Sanity check the magnitude. Between them they close 15 ft for every one of Bella's 2.5-ft steps, so Bella personally covers only

\frac{2.5}{15}

\frac{1}{6}

of the trip — about

\frac{10,560}{6}

= 1,760 ft, or one-third of a mile. At 2.5 ft per step that is

\frac{1760}{2.5}

= 704 steps, matching (A). It also matches the ratio intuition: if Ella is 5× faster, Bella covers

\frac{1}{1+5}

\frac{1}{6}

of the gap. Choice (D) 1760 is the trap that confuses "feet Bella walks" with "steps Bella takes," and choice (E) 3520 is the trap of using half the distance (1 mile) without the speed ratio.

💡Key takeaway

This AMC 8 problem only needs Grade 5 decimal arithmetic and "times as fast" comparison you already know!

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