AMC 8 · 2016 · #16

Grade 7 rate-ratioalgebra

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up

rateratio-proportionpercentagelinear-equations-one-var convert-to-algebrapattern-recognition ↑ Prerequisites: ratio-proportionpercentage

📏 Medium solution 💡 3 insights

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Problem

Annie and Bonnie start together on a 400-meter oval track. Annie runs 25% faster than Bonnie. Annie first passes (laps) Bonnie when she has gone exactly one full lap more than Bonnie. How many laps has Annie run at that moment?

Pick an answer.

(A)

$1\dfrac{1}{4}$

(B)

$3\dfrac{1}{3}$

(C)

(D)

(E)

AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Find a Pattern

The key insight is a ratio pattern: when two runners go for the same time, the ratio of their distances equals the ratio of their speeds (Tool #11, Find a Pattern). Turning "25% faster" into the clean ratio 5:4 exposes that pattern. Tool #4 (Use a Variable) lets us call Bonnie's lap count L_B and Annie's L_A, and the lapping condition becomes the equation L_A = L_B + 1. Tool #13 (Solve an Equivalent Problem) reframes the geometry: the 400 m track length never enters the calculation — "first passes" is equivalent to "the gap between them equals exactly 1 lap." Working in lap counts instead of meters cancels the track length entirely.

1STEP 1

"25% faster" means Annie's speed is $\frac{5}{4}$ of Bonnie's, a speed ratio of 5 : 4.

\frac{v_A}{v_B}

= 1.25 =

\frac{5}{4}

2STEP 2

Equal running time makes the lap-count ratio equal the speed ratio: $\frac{L_A}{L_B}$ = $\frac{5}{4}$ .

\frac{L_A}{L_B}

\frac{v_A}{v_B}

\frac{5}{4}

3STEP 3

On a loop, "first passes" means Annie ran one more lap: L_A = L_B + 1.

L_A = L_B + 1

4STEP 4

Substitute the ratio into L_A = L_B + 1 and solve: L_B = 4.

\frac{5}{4}

L_B = L_B + 1 →

\frac{1}{4}

L_B = 1 → L_B = 4

5STEP 5

Add one more lap for Annie: L_A = 4 + 1 = 5 → (D).

L_A = L_B + 1 = 4 + 1 = 5 → (D)

Answer

Per lap by Bonnie, Annie runs

\frac{5}{4}

of a lap, so Annie's lead grows by

\frac{1}{4}

lap every time Bonnie completes one lap. To open up a full 1-lap lead, Bonnie needs 4 laps and Annie runs 5 laps. The numbers are small, integer, and consistent with the answer choices — and choice (D) = 5 matches.

💡Key takeaway

This AMC 8 problem only needs Grade 7 algebra — a speed ratio plus a one-step linear equation — that you already know!

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