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2017 AMC 10B Problem 2

2017 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: October 28, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

Sofia ran $5$ laps around the $400$ -meter track at her school. For each lap, she ran the first $100$ meters at an average speed of $4$ meters per second and the remaining $300$ meters at an average speed of $5$ meters per second. How much time did Sofia take running the $5$ laps?

A) $\text{5 minutes and 35 seconds}$
B) $\text{6 minutes and 40 seconds}$
C) $\text{7 minutes and 5 seconds}$
D) $\text{7 minutes and 25 seconds}$
E) $\text{8 minutes and 10 seconds}$

Answer

Key insight

Find the time for each part of a lap: 100 meters takes 25 seconds and 300 meters takes 60 seconds, so one lap is 85 seconds.

Solution

Speeds cannot simply be averaged, but times can be added. So find the time for each part of a lap, then add.

First $100$ meters at $4$ meters per second: $100 \div 4 = 25$ seconds.
Remaining $300$ meters at $5$ meters per second: $300 \div 5 = 60$ seconds.
One lap takes $25 + 60 = 85$ seconds.
Five laps take $5 \cdot 85 = 425$ seconds.
Change to minutes: $7$ minutes is $420$ seconds, and $425 - 420 = 5$ . So the time is $7$ minutes and $5$ seconds.

The answer is $\boxed{\textbf{(C)}\ \text{7 minutes and 5 seconds}}$ .

Why this works

Whenever a trip is made of pieces at different speeds, work out each piece's time with time $=$ distance $\div$ speed, and add the times. Leave the change from seconds to minutes for the very end, and check it, because that is where careless slips happen.

The trap

Averaging the two speeds to 4.5 meters per second. The two parts have different lengths, so a plain average of the speeds gives the wrong time.

Concise solution

Compute the time for one lap segment by segment.

First $100$ meters at $4$ m/s: $\frac{100}{4} = 25$ seconds.
Remaining $300$ meters at $5$ m/s: $\frac{300}{5} = 60$ seconds.

One lap takes $25 + 60 = 85$ seconds, so five laps take $5 \cdot 85 = 425$ seconds. Since $425 = 7 \cdot 60 + 5$ , that is $7$ minutes and $5$ seconds.

The answer is $\boxed{\textbf{(C)}\ \text{7 minutes and 5 seconds}}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)

Techniques

Set up the equation/formula and compute; no special trick needed