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2003 AMC 10A Problem 4

2003 AMC 10A Problem 4AMC 10 Step by Step

Name: Date: September 13, 2026

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

It takes Anna $30$ minutes to walk uphill $1$ km from her home to school, but it takes her only $10$ minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?

A)
$3$
B)
$3.125$
C)
$3.5$
D)
$4$
E)
$4.5$

Answer

Key insight

Average speed is total distance over total time: 2 km in 40 minutes, which is 3 km/hr.

Solution

The round trip covers $1 + 1 = 2$ km and takes $30 + 10 = 40$ minutes, which is $\tfrac{40}{60} = \tfrac{2}{3}$ of an hour.

Average speed is total distance divided by total time:
$\frac{2}{2/3} = 3 \text{ km/hr}.$

The answer is $\boxed{\textbf{(A)}\ 3}$ .

Why this works

Average speed is never the average of the speeds unless the times are equal. Here Anna spends three times as long on the slow leg, so the slow speed dominates. Always go back to the definition: total distance over total time.

Alternative approach

For equal distances at speeds $u$ and $v$ , the average speed is the harmonic mean $\frac{2uv}{u+v}$ . With $u = 2$ and $v = 6$ km/hr this is $\frac{24}{8} = 3$ .

The trap

Averaging the two speeds 2 km/hr and 6 km/hr to get 4, choice (D).

Common mistakes

Averaging the two speeds 2 km/hr and 6 km/hr to get 4, choice (D).
Forgetting to convert $40$ minutes to hours and reporting $2/40$ or similar.

Techniques

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