2025 AMC 8 Problem 3

One Page to Learn. One Point to Earn.

In This Page:

To help you master 2025 AMC 8 Problem 3, 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 3

Solution

Solution 1

Total Quantity Method

Since the same deck of cards is used in both games, the total number of cards stays the same. We can calculate this total using the information from the first game, and then divide it equally among the larger group of players in the second game.

Total = (Number of Groups) × (Size of Each Group). Here, the groups are the players, and the size of each group is the number of cards they receive.

Step 1: Find the initial number of players

The game involves Annika and her friends. We must count Annika as a player along with her friends to find the starting number of people.

$$1+3=4\text{players}$$

Step 2: Find the total number of cards

Since the cards are distributed evenly, we multiply the number of players by the number of cards each player receives to find the total number of cards in the deck.

$$4\times 15=60\text{total cards}$$

Step 3: Find the new number of players

For the next game, two more friends join the original group.

$$4+2=6\text{players}$$

Step 4: Calculate the new number of cards per player

The total cards are now distributed evenly among the new number of players. We divide the constant total number of cards by this new number.

$$\frac{60}{6}=10\text{cards per player}$$

Correct answer: C. $10$
The answer is 10 cards per player, which corresponds to Choice C. To check, 6 players each holding 10 cards gives 60 total cards. 4 players each holding 15 cards also gives 60 total cards. The totals match perfectly.

Solution 2

Proportional Reasoning Method

Because the total number of cards does not change, if you increase the number of players by a certain multiplier, the number of cards each player receives must be multiplied by the reciprocal of that multiplier.

Inverse Variation: When the product of two changing quantities is constant, they are inversely proportional. Multiplying one quantity by a factor $k$ means multiplying the other by $\frac{1}{k}$.

Step 1: Identify the constant relationship

The total number of cards is constant across both games. This implies that the number of players and the number of cards per player are inversely proportional.

$$\text{Players}\times \text{Cards per Player}=\text{Constant Total}$$

Step 2: Determine the multiplier for the number of players

Calculate the ratio of the new number of players to the original number of players. The original group had 4 players, and the new group has 6 players.

$$\text{Ratio}=\frac{6}{4}=\frac{3}{2}$$

Step 3: Apply the inverse multiplier to the cards per player

Since the number of players was multiplied by a factor, the number of cards per player must be multiplied by the reciprocal of that factor to keep the total product the same.

$$15\times \frac{2}{3}=10\text{cards per player}$$

Correct answer: C. $10$
The answer is 10 (Choice C). Checking the proportions: the number of players increased by a factor of 1.5, so the number of cards per player should be divided by 1.5. $15÷1.5=10$, which confirms our result.

Similar Problems in Real Exams

The following five AMC 8 problems are closely related to AMC 8 2025 Question 3. These real AMC 8 problems test the same key concept—Arithmetic Operations & Number Sense. Click any problem to learn and practice, then continue to the adapted practice section later in this article for more problems to try.

2026 AMC 8 Problem 1

2025 AMC 8 Problem 2

2023 AMC 8 Problem 1

2023 AMC 8 Problem 3

2020 AMC 8 Problem 3

Adapted Problems to Practice

What Formulas Do You Need for This Problem?

• Total Quantity for Equal Groups:

The Number of Groups represents the players, and the Size of Each Group represents the cards per player. This applies whenever a resource is shared evenly. Used to find the fixed size of the deck when we know how many cards each of the 4 players received.

• Inverse Variation:

Applies to situations where a total amount is fixed but divided into variable numbers of parts. Since the total deck of cards remains the same, an increase in the number of players by a factor of $\frac{3}{2}$ strictly forces the cards per player to change by a factor of $\frac{2}{3}$.

Key Takeaway: Remember This Strategy for Next Time

Unlock More Targeted AMC 8 Practice Resource

Build Stronger Skills. Practice with Purpose. Prepare with Confidence.

Get a complete set of module-focused AMC 8 practice workbooks designed to help students strengthen core concepts, improve problem-solving strategies, and feel more confident on test day.

With 250 practice problems organized across six essential AMC 8 modules, students can focus on one skill area at a time and learn from clear, step-by-step solutions.

Download the Free AMC 8 Practice Resource Pack

Inside, You’ll Find:

• 6 module-focused practice workbooks: Algebra, Geometry, Number Theory, Combinatorics, and Probability & Statistics.
• 250 AMC 8-style practice problems covering essential concepts and common competition question types.
• Complete answer keys and step-by-step solutions to help students understand the reasoning behind each answer.
• Organized, print-ready materials for independent study, guided practice, or extra preparation at home.

Get Instant Access: Fill out the form below to receive our exclusive resource pack.

*Please enter a valid email address to access the materials and receive future prep updates.

About Think Academy

Think Academy US Online, part of TAL Education Group (NYSE: TAL), provides structured math and ELA programs that help students build strong academic foundations and grow with confidence. With expert teachers, proven methods, and innovative AI tools, we nurture learning ability, curiosity, and healthy study habits that support long-term growth and a lifelong love of learning.

Check our programs:

• Think Kids: Ages 3-5
• Math
  • Lower Elementary Math — Grades 1–2
  • Upper Elementary Math — Grades 3–5
  • Middle School Math — Grade 6–8
  • High School Math — Grade 9-12
• ELA
  • Elementary ELA — Grades 2–5

Want more insights on Math & ELA learning and parenting? Fill out the form on this page and subscribe to our newsletter for weekly tips and the latest resources. Join 100,000+ families in our Parent Community!