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2010 AMC 10B Problem 10

2010 AMC 10B Problem 10AMC 10 Step by Step

Name: Date: September 13, 2026

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

Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain?

A)
$18$
B)
$21$
C)
$24$
D)
$27$
E)
$30$

Answer

Key insight

Work in miles per minute: 1/2 mile per minute in sun and 1/3 in rain; with t rain minutes, (40 - t)/2 + t/3 = 16.

Solution

Convert the speeds to miles per minute: $30$ mph is $\frac{1}{2}$ mile per minute and $20$ mph is $\frac{1}{3}$ mile per minute.

Let $t$ be the number of minutes in the rain, so she spent $40 - t$ minutes in the sun. Total distance:
$\frac{40 - t}{2} + \frac{t}{3} = 16.$
Multiply by $6$ : $3(40 - t) + 2t = 96$ , so $120 - t = 96$ and $t = 24$ .

Check: $16$ sunny minutes cover $8$ miles and $24$ rainy minutes cover $8$ miles, total $16$ miles.

The answer is $\boxed{\textbf{(C)}\ 24}$ .

Why this works

Two-rate trips are a single linear equation once the unknown is chosen well: let the unknown be the quantity asked for, express the other time as the remainder, and add the distances. Converting rates to the same time unit as the data (minutes) avoids fractions of an hour.

Alternative approach

If all $40$ minutes were in the sun she would cover $20$ miles, $4$ too many. Each minute switched to rain loses $\frac12 - \frac13 = \frac16$ mile, so $4 \cdot 6 = 24$ minutes must be rainy.

The trap

Mixing hours and minutes, e.g. using 40 as the total time in hours, or solving for the sunny minutes (16) and stopping.

Common mistakes

Mixing hours and minutes, e.g. using 40 as the total time in hours, or solving for the sunny minutes (16) and stopping.
Averaging the two speeds to $25$ mph and assuming equal times, which does not use the $16$ -mile constraint.

Techniques

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