Question

The one-way routes connecting towns $A,$ $M,$ $C,$ $X,$ $Y,$ and $Z$ are shown in the figure below (not drawn to scale). The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from $A$ to $Z$ in kilometers?

$28$

$29$

$30$

$31$

$32$

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Written solution:

A systematic way of tracking the shortest overall distance to $Z$ is to consider the shortest distance to get to each town from $A$ . For instance, the shortest distance to get to town $X$ from $A$ is $5$ km, trivially.

Then, for town $M$ , going to town $X$ first will be shorter compared to going directly from $A$ , so the shortest path to town $M$ has a length of $7$ km.

For town $Y$ , it will take us $15$ km if we come from town $X$ and only $13$ km coming from $M,$ so $13$ km is the length of shortest path to $Y$ from $A$ .

Doing the same for town $C$ will give us $18$ km as the shortest distance by coming from town $Y$ .

Finally, for town $Z$ , we can either come from town $Y,$ $C,$ or $M$ . The total distance if we come from each three towns respectively would be $30,$ $28,$ and $32$ . Hence, $28$ km is the shortest distance from $A$ to $Z$ .

Thus, A is the correct answer.

2024 AMC 8 Problem 14

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Problem 14 in Other Years