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

📘 View easy version →

Problem

Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely 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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

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

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

#8 Analyze the Units 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.

$$\text{rate} = \dfrac{6 \text{ cars}}{10 \text{ s}} = 0.6 \tfrac{\text{cars}}{\text{s}}$$

💡 Dividing a count by the time it took is the Grade 6 definition of a unit rate.

#8 Analyze the Units 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.

$$2 \text{ min } 45 \text{ s} = 2 \times 60 + 45 = 120 + 45 = 165 \text{ s}$$

💡 Converting minutes to seconds within the same time system is the Grade 5 measurement-conversion standard.

#8 Analyze the Units 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.

$$\text{total cars} = 0.6 \tfrac{\text{cars}}{\text{s}} \times 165 \text{ s} = \tfrac{3}{5} \times 165 = 3 \times 33 = 99 \text{ cars}$$

💡 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.

#7 Identify Subproblems 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 $100$.

$$99 \approx 100 \;\Rightarrow\; \textbf{(C)}$$

💡 On AMC problems that say "most likely," you compute the model number and round to the nearest answer choice.

[1] #8 6.RP.A.2 Find the rate of cars per second. Trey saw $6$ cars in $10$ seconds, so divide c

[2] #8 5.MD.A.1 Convert the total clearing time into seconds so it matches the rate's unit. $2$

[3] #8 6.RP.A.3 Multiply rate by total time. The seconds in the denominator cancel against the s

[4] #7 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.1 Convert 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.2 Understand the concept of a unit rate (Turning "$6$ cars in $10$ seconds" into the unit rate $0.6$ cars per second.)
6.RP.A.3 Use 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

More like this