AMC 8 · 2013 · #7

Grade 6 rate-ratioarithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

rateunit-conversionestimation dimensional-analysisidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Medium solution 💡 3 insights

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Problem

A train passes a crossing at a constant speed. In the first 10 seconds Trey counts 6 cars. The whole train takes 2 minutes 45 seconds to clear the crossing. Of the given choices, which is closest to the total number of cars in the train?

Pick an answer.

(A)

(B)

(C)

100

(D)

120

(E)

140

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

This is a rate problem in disguise: cars per second is a unit rate, and the answer is (rate) × (total time). Tool #8 (Analyze the Units) keeps the bookkeeping clean — we need cars-per-second on one side and seconds on the other so the "seconds" cancel and only "cars" are left. Tool #7 (Identify Subproblems) splits the work into three small pieces: (1) find the rate, (2) convert 2 min 45 s into a single time unit, (3) multiply. Solving each piece on its own is much easier than handling the whole thing at once.

1STEP 1

Divide the count by the time: 6 cars in 10 seconds gives a unit rate of 0.6 cars per second.

rate = (6 cars)/(10 s) = 0.6 cars/s

2STEP 2

Match units: 2 minutes is 2 × 60 = 120 seconds, plus 45 more makes 165 seconds in all.

2 min 45 s = 2 × 60 + 45 = 120 + 45 = 165 s

3STEP 3

Multiply rate by time; the seconds cancel and leave cars: 0.6 × 165 = $\frac{3}{5}$ × 165 = 99 cars.

total cars = 0.6 cars/s × 165 s =

\frac{3}{5}

× 165 = 3 × 33 = 99 cars

4STEP 4

The choices are 60, 80, 100, 120, 140; 99 sits just one away from 100, so that is the closest.

99 ≈ 100 → (C)

Answer

100

Sanity-check with a rougher estimate: 6 cars in 10 s is the same as 36 cars per minute. The train passes for almost 3 full minutes (2 min 45 s), so roughly 36 × 3 = 108 cars — but the actual time is

\frac{1}{4}

minute short of 3 minutes, so subtract about 36 ×

\frac{1}{4}

= 9 cars to get ∼ 99 cars. That matches the exact answer and is much closer to 100 than to any other choice.

💡Key takeaway

Counting how many things pass in a few seconds gives you a per-second rate; multiply by the total time and you get the whole count — a Grade 6 rate-reasoning trick.

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