AMC 8 · 2013 · #7
Easy mode Grade 6Problem
Trey and his mom stopped at a railroad crossing while a train rolled past. The train moved at a steady speed the whole time.
When the train started passing, Trey counted the cars. In the first 10 seconds, 6 cars went by.
The whole train took 2 minutes and 45 seconds to finish crossing.
Which answer is closest to the most likely total number of cars in the train?
Pick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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?
Givens: $6$ cars pass in the first $10$ seconds; Train speed is constant; Total clearing time $= 2$ min $45$ s; Answer choices: (A) $60$, (B) $80$, (C) $100$, (D) $120$, (E) $140$
Unknowns: The total number of cars in the train (closest choice)
Understand
Restated: 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?
Givens: $6$ cars pass in the first $10$ seconds; Train speed is constant; Total clearing time $= 2$ min $45$ s; Answer choices: (A) $60$, (B) $80$, (C) $100$, (D) $120$, (E) $140$
Plan
Primary tool: #8 Analyze the Units
Secondary: #7 Identify Subproblems
This is a rate problem in disguise: cars per second is a unit rate, and the answer is (rate) $\times$ (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.
Execute — Answer: C
6.RP.A.2 Step 1 - Find the rate of cars per second.
- Trey saw $6$ cars in $10$ seconds, so divide cars by seconds to get a unit rate.
💡 Dividing a count by the time it took is the Grade 6 definition of a unit rate.
5.MD.A.1 Step 2 - Convert the total clearing time into seconds so it matches the rate's unit.
- $2$ minutes is $2 \times 60 = 120$ seconds, plus the extra $45$ seconds.
💡 Converting minutes to seconds within the same time system is the Grade 5 measurement-conversion standard.
6.RP.A.3 Step 3 - Multiply rate by total time.
- The seconds in the denominator cancel against the seconds of the time, leaving cars.
💡 Multiplying a unit rate by an amount of time to get a total is Grade 6 rate reasoning. Rewriting $0.6$ as $\tfrac{3}{5}$ makes the arithmetic exact.
6.RP.A.3 Step 4 - Match the computed total to the closest answer choice.
- The five choices are $60, 80, 100, 120, 140$, and $99$ is one away from $100$, so the closest choice is $100$.
💡 On AMC problems that say "most likely," you compute the model number and round to the nearest answer choice.
6.RP.A.2 Find the rate of cars per second. Trey saw $6$ cars in $10$ seconds, so divide c 5.MD.A.1 Convert the total clearing time into seconds so it matches the rate's unit. $2$ 6.RP.A.3 Multiply rate by total time. The seconds in the denominator cancel against the s 6.RP.A.3 Match the computed total to the closest answer choice. The five choices are $60, Review
Reasonableness: 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 $36 \times 3 = 108$ cars — but the actual time is $\tfrac{1}{4}$ minute short of $3$ minutes, so subtract about $36 \times \tfrac{1}{4} = 9$ cars to get $\sim 99$ cars. That matches the exact answer and is much closer to $100$ than to any other choice.
Alternative: Tool #6 (Guess and Check) on the choices: if the train has $N$ cars and takes $165$ s, the rate is $\tfrac{N}{165}$ cars/s, which should equal the observed $0.6$ cars/s. Solving $\tfrac{N}{165} = 0.6$ gives $N = 99$, and $99$ is closest to choice (C) $100$.
CCSS standards used (min grade 6)
5.MD.A.1Convert among different-sized standard measurement units within a given system (Converting $2$ minutes $45$ seconds into a single unit of seconds ($165$ s) so the rate and time share units.)6.RP.A.2Understand the concept of a unit rate (Turning "$6$ cars in $10$ seconds" into the unit rate $0.6$ cars per second.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Multiplying the unit rate by the total clearing time to estimate the total number of cars ($0.6 \times 165 = 99$) and matching it to the nearest answer choice.)
⭐ 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.
⭐ 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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