Question

Two towns, $A$ and $B,$ are connected by a straight road, $15$ miles long. Traveling from town $A$ to town $B,$ the speed limit changes every $5$ miles: from $25$ to $40$ to $20$ miles per hour (mph). Two cars, one at town $A$ and one at town $B,$ start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town $A,$ in miles, will the two cars meet?

$7.75$

$8$

$8.25$

$8.5$

$8.75$

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Written solution:

Think of the road as having three sections: left, middle, and right. Each section is $5$ miles long.

The car from $A$ reaches the middle section in $\frac{5}{25} = \frac{1}{5}$ hours.

The car from $B$ reaches the middle section in $\frac{5}{20} = \frac{1}{4}$ hours. By that time, the car from $A$ has already driven in the middle section for $\frac{1}{4} - \frac{1}{5} = \frac{1}{20}$ hours. During this time, that car from $A$ has traveled $40 \times \frac{1}{20} = 2$ miles in the middle section.

That leaves $5 - 2 = 3$ miles between the two cars in the middle section.

At that moment, the car from $A$ is $5 + 2 = 7$ miles from $A.$

Since the cars drive at the same speed of $40$ mph in the middle section, they meet after each driving $1.5$ more miles. This takes the car from $A$ a total distance of $7 + 1.5 = 8.5$ miles, which is choice D.

2025 AMC 8 Problem 19

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Problem 19 in Other Years