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 seconds, cars went by.
The whole train took minutes and seconds to finish crossing.
Which answer is closest to the most likely total number of cars in the train?
Pick an answer.
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.