2025 AMC 8 Problem 17

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To help you master 2025 AMC 8 Problem 17, 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

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2025 AMC8 Problem 17

Solution

Solution 1

Direct Component Calculation Method

The people who work in city A come from three specific groups: commuters from city B, commuters from city C, and residents of city A who stay in their own city. We can calculate the size of each group separately and add them together.

Step 1: Calculate the number of commuters from city B to city A

The diagram shows that a fraction of the people living in city B commute to city A. We multiply the population of city B by this fraction.

120 \times \frac{1}{3} = 40

Step 2: Calculate the number of commuters from city C to city A

Similarly, a fraction of the people living in city C commute to city A. We multiply the population of city C by this fraction.

160 \times \frac{1}{8} = 20

Step 3: Calculate the number of people who live and work in city A

The problem states everyone works in one of the three cities. The people living in city A who do not commute to B or C must stay and work in A. First, we find the number of people leaving A. Then, we subtract this from the total population of A.

Commuters to B = 100 \times \frac{1}{4} = 25

Commuters to C = 100 \times \frac{1}{5} = 20

Total leaving A = 25 + 20 = 45

Staying in A = 100 - 45 = 55

Step 4: Calculate the total number of people working in city A

Now we add the three groups of people who work in city A: those coming from B, those coming from C, and those staying in A.

Total workers in A = 40 + 20 + 55 = 115

Correct answer: D. $115$
The total number of people working in A is 115, which corresponds to answer choice D. We can verify our ‘staying in A’ calculation by working with fractions instead:

1 - (1/4 + 1/5) = 1 - 9/20 = 11/20

, and

100 \times (11/20) = 55

. This confirms our integer subtraction was correct.

Solution 2

Net Flow Method

Instead of explicitly calculating how many people stay in city A, we can treat the population of A as a starting base. We add everyone who commutes into A (the inflow) and subtract everyone who commutes out of A (the outflow).

Step 1: Calculate the total inflow to city A

The inflow consists of all people commuting into city A from the other two cities.

Inflow from B = 120 \times \frac{1}{3} = 40

Inflow from C = 160 \times \frac{1}{8} = 20

Total Inflow = 40 + 20 = 60

Step 2: Calculate the total outflow from city A

The outflow consists of all people living in city A who commute to other cities.

Outflow to B = 100 \times \frac{1}{4} = 25

Outflow to C = 100 \times \frac{1}{5} = 20

Total Outflow = 25 + 20 = 45

Step 3: Apply the net flow equation

The total number of people working in city A is the original number of residents, plus the people who enter, minus the people who leave.

Workers in A = Residents of A + Total Inflow - Total Outflow

Workers in A = 100 + 60 - 45

Workers in A = 160 - 45 = 115

Correct answer: D. $115$
The result is 115, matching choice D. This matches our previous method perfectly and demonstrates how treating the problem as a single system of ‘ins’ and ‘outs’ gives the exact same result with a genuinely different logical perspective.

Adapted Problems to Practice

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What Formulas Do You Need for This Problem?

Inflow-Outflow Principle:

Final Amount = Initial Amount + Inflow - Outflow

Applies to any system where objects, people, or quantities move between discrete categories or locations without being created or destroyed. People commute between cities, but the total number of people remains constant. We can track the net change in a single city's working population during the day.

Key Takeaway: Remember This Strategy for Next Time

A problem asks for a remaining portion after multiple fractions are removed (like finding the people who stay in city A).

Try: Calculate the exact whole numbers for the removed parts first, then subtract from the total, instead of trying to add and subtract fractions of a whole.

Why: Working with whole numbers is usually faster and less prone to arithmetic mistakes than finding common denominators for fractions.

A problem involves people or objects moving back and forth between different groups.

Try: Consider using the Inflow-Outflow principle: Final = Initial + Inflow - Outflow.

Why: This strategy lets you focus entirely on the movement across borders. It allows you to skip calculating the exact number of objects that stay stationary inside their original group.

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2025 AMC 8 Problem 17

In This Page:

2025 AMC8 Problem 17