Printable

2009 AMC 10A Problem 20

2009 AMC 10A Problem 20AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 3 of 5 · about 3 min · position P16-20

Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?

A)
$20$
B)
$30$
C)
$55$
D)
$65$
E)
$80$

Answer

Key insight

The 1 km/min closing rate splits 3:1, so Lauren rides 1/4 km/min; after 5 minutes 15 km remain and Lauren alone needs 60 more minutes.

Solution

Let Lauren's speed be $v$ km/min; Andrea's is $3v$ . Approaching each other, the gap closes at $v + 3v = 4v = 1$ km/min, so
$v = \tfrac{1}{4} \text{ km/min}, \qquad 3v = \tfrac{3}{4} \text{ km/min}.$

In the first $5$ minutes the gap shrinks by $5 \cdot 1 = 5$ km, leaving $20 - 5 = 15$ km between them.

After Andrea stops, only Lauren moves, at $\frac14$ km/min. Covering $15$ km takes
$\frac{15}{1/4} = 60 \text{ minutes}.$

Total time from the start: $5 + 60 = 65$ minutes.

The answer is $\boxed{\textbf{(D)}\ 65}$ .

Why this works

The closing rate is the sum of the two speeds, and a $3:1$ speed ratio splits that rate into $\frac34$ and $\frac14$ . The problem has two phases with different rates; track the remaining distance at the phase change, then apply distance $=$ rate $\times$ time once more. Note that Andrea's individual speed is never needed after the split.

The trap

Forgetting that Andrea stops and using the combined closing rate for the whole 20 km, which gives 20 minutes.

Common mistakes

Forgetting that Andrea stops and using the combined closing rate for the whole 20 km, which gives 20 minutes.
Answering $60$ (the second phase only) and forgetting to add the first $5$ minutes, or using Andrea's speed $\frac34$ for Lauren.

Techniques

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