2025 AMC 8 Problem 5

Question

Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory (labeled

F

) and drives to location

A,

then

B,

then

C,

before returning to

F.

What is the shortest distance, in blocks, she can drive to complete the route?

20

22

24

26

28

Answer: C

Concepts:coordinate geometry optimization

Difficulty rating: 960

Small hint:

Break the route into four shortest grid trips

Big hint:

On a street grid, count horizontal blocks plus vertical blocks

Video solution:

Click to load, then click again to play

Written solution:

The key idea is that if driving from coordinates

(x_1, y_1)

to

(x_2, y_2),

then the shortest distance is

|x_2 - x_1| + |y_2 - y_1|.

This is often called the Manhattan distance. It is also equal to the number of horizontal blocks between the locations, plus the number of vertical blocks between the locations.

The shortest distance from

F

to

A

is then

1 + 2 = 3.

The shortest distance from

A

to

B

is

7 + 3 = 10.

The shortest distance from

B

to

C

is

2 + 4 = 6.

The shortest distance from

C

to

F

is

4 + 1 = 5.

Adding up all of these numbers, we get

3 + 10 + 6 + 5 = 24,

which is choice C.

One small possible shortcut for the solution is to notice that when going from

B

to

C

to

F,

the visit to

C

is conveniently along a shortest path from

B

to

F

anyway, so we can even remove the requirement to stop at

C

from the problem.

2025 AMC 8 Problem 5

Answer: C

Difficulty rating: 960

Small hint:

Big hint:

Video solution:

Written solution:

Problem 5 in other years