2025 AMC 8 Problem 5

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In This Page:

To help you master 2025 AMC 8 Problem 5, this page includes:

• The original AMC problem to try, plus the correct answer

• Clear, step-by-step solutions using different methods

• Key ideas, useful formulas, and takeaways you can apply to similar problems

• Related problems from past AMC exams, plus new practice problems that build the same skills

Want more practice? Scroll down to download Think Academy’s AMC 8 practice packs and keep building your AMC 8 skills.

2025 AMC8 Problem 5

Solution

Solution 1

Stage-by-Stage Counting Method

The shortest path between two points on a grid is found by adding the horizontal and vertical blocks between them. We can find the length of each leg of the journey and add them together.

Grid Distance: The shortest distance traveling along a grid between two points is the number of horizontal steps plus the number of vertical steps.

Step 1: Calculate the distance from F to A

To get from location F to location A, Betty must drive right and down. By counting the blocks on the grid, she moves right by 1 block and down by 2 blocks.

$${\text{Distance}}_{F\to A}=1+2=3\text{blocks}$$

Step 2: Calculate the distance from A to B

From location A to location B, Betty must drive left and down. Counting on the grid, she moves left by 7 blocks and down by 3 blocks.

$${\text{Distance}}_{A\to B}=7+3=10\text{blocks}$$

Step 3: Calculate the distance from B to C

From location B to location C, Betty must drive right and up. She moves right by 2 blocks and up by 4 blocks.

$${\text{Distance}}_{B\to C}=2+4=6\text{blocks}$$

Step 4: Calculate the distance from C to F

Finally, to return from location C to the starting point F, she drives right and up. She moves right by 4 blocks and up by 1 block.

$${\text{Distance}}_{C\to F}=4+1=5\text{blocks}$$

Step 5: Calculate the total route distance

Add the distances of all four stages together to find the total distance driven.

$$\text{Total Distance}=3+10+6+5$$

$$\text{Total Distance}=24\text{blocks}$$

Correct answer: C. $24$
The shortest total distance is 24 blocks, which corresponds to answer choice C. $\boxed{24}$

Solution 2

Coordinate Geometry Method

By placing the grid on a coordinate plane, we can assign an $(x,y)$ pair to each location. We can then calculate the grid distances using arithmetic rather than visual counting.

Manhattan Distance Formula: The distance between $(x_1,y_1)$ and $(x_2,y_2)$ on a grid is $|x_1-x_2|+|y_1-y_2|$.

Step 1: Set up a coordinate system

Let the bottom-left corner of the map, location B, be the origin.

$$B=(0,0)$$

Step 2: Find the coordinates of the other locations

By counting the grid lines starting from B, we determine the horizontal ($x$) and vertical ($y$) positions of locations C, F, and A.

$$C=(2,4)$$

$$F=(6,5)$$

$$A=(7,3)$$

Step 3: Calculate the distance from F to A

Using the coordinates of F and A, find the absolute difference in their x-coordinates and y-coordinates.

$$d(F,A)=|6-7|+|5-3|$$

$$d(F,A)=|-1|+|2|=1+2=3$$

Step 4: Calculate the distance from A to B

Repeat the process for the journey from A to B.

$$d(A,B)=|7-0|+|3-0|$$

$$d(A,B)=|7|+|3|=7+3=10$$

Step 5: Calculate the distance from B to C

Repeat the process for the journey from B to C.

$$d(B,C)=|0-2|+|0-4|$$

$$d(B,C)=|-2|+|-4|=2+4=6$$

Step 6: Calculate the distance from C to F

Repeat the process for the final leg from C to F.

$$d(C,F)=|2-6|+|4-5|$$

$$d(C,F)=|-4|+|-1|=4+1=5$$

Step 7: Calculate the total route distance

Sum the distances of all four stages to find the total distance.

$$\text{Total}=3+10+6+5=24$$

Correct answer: C. $24$
The total distance calculated using coordinates perfectly matches our visual counting method. The result is $\boxed{24}$.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2025 Question 5. These real AMC 8 problems test the same key concept—Coordinate Shapes & Loci. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.

2024 AMC 8 Problem 11

2022 AMC 8 Problem 18

2015 AMC 8 Problem 19

2006 AMC 8 Problem 5

2002 AMC 8 Problem 20

Adapted Problems to Practice

What Formulas Do You Need for This Problem?

• Taxicab Geometry (Manhattan Distance):

$d$ is the shortest grid distance between points $(x_1,y_1)$ and $(x_2,y_2)$. Because the truck must drive along the street grid, it cannot take diagonal shortcuts. Thus, we must calculate the total horizontal and vertical distance rather than a straight-line (Euclidean) distance.

Key Takeaway: Remember This Strategy for Next Time

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