Competition · AMC preparation · step 4 of 4
AMC 8 · 2016 · #16
Grade 7 rate-ratioalgebraPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Turn the percent into a ratio
"25% faster" means Annie's speed is of Bonnie's, a speed ratio of 5 : 4.
Converting "25% more" to the fraction 5/4 is the Grade 6 percent-as-ratio move.
6.RP.A.3Work BackwardsMatch distance ratio to speed ratio
Equal running time makes the lap-count ratio equal the speed ratio: = .
Equal time → distances scale exactly with speeds — that is the rate pattern (Tool #11).
Because Annie and Bonnie run for the same amount of time, the number of laps each has run is in the same ratio as their speeds, 5 : 4.
▸ Why?
In that one shared time, each girl's distance is her speed repeated over the same stretch of time, so Annie piles up v_A time-lengths and Bonnie piles up v_B, and the two distances line up in the very same pattern as v_A : v_B.
▸ Why?
Both distances are built from equal-size pieces of the same length, so comparing the totals is just comparing how many pieces each one has, which is v_A against v_B.
▸ Why?
Measured in meters, each girl's distance is her lap count times the same 400 m per lap, so the distance ratio is (L_A · 400) : (L_B · 400) — the identical factor 400 sits in both the top and the bottom.
▸ Why?
Dividing the top and the bottom of (L_A · 400)/(L_B · 400) by the shared 400 leaves the value unchanged, so it collapses to L_A/L_B — the lap ratio is exactly the distance ratio.
Write the lapping equation
On a loop, "first passes" means Annie ran one more lap: L_A = L_B + 1.
Naming the two lap counts as variables turns the word condition into a clean equation (Tool #4).
6.EE.B.7Introduce A VariableSolve the system
Substitute the ratio into L_A = L_B + 1 and solve: L_B = 4.
A one-step substitution leaves a linear equation in one variable — Grade 7 algebra.
7.EE.B.4Introduce A VariableRead off Annie's laps
Add one more lap for Annie: L_A = 4 + 1 = 5 → (D).
The 400 m track length never appeared — the answer depends only on the lap-count gap (Tool #13, equivalent problem).
6.RP.A.3Convert To AlgebraThis AMC 8 problem only needs Grade 7 algebra — a speed ratio plus a one-step linear equation — that you already know!
- Turn the percent into a ratio
- Match distance ratio to speed ratio
- Write the lapping equation
- Solve the system
- Read off Annie's laps
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