Question

A, B, C, D, and E play a game of cards. A says to B, "If you give me three cards, you will have as many as E has and if I give you three cards, you will have as many as D has." A and B together have 10 cards more than what D and E together have. If B has two cards more than what C has and the total number of cards be 133, how many cards does B have?

• A. 35
• B. 26
• C. 33
• D. 25
• E.

Hint:

Use a system of linear equations to solve problems involving relationships between quantities, and substitute known values to find the unknowns.

Answer 1

The Correct Option is D

Step 1: Define Variables Let \( A, B, C, D, E \) represent the number of cards each person has. Step 2: Translate the Statements into Equations - Statement 1: "If you give me three cards, you will have as many as E has." \[ B - 3 = E \quad \text{(1)} \] - Statement 2: "If I give you three cards, you will have as many as D has." \[ B + 3 = D \quad \text{(2)} \] - Statement 3: "A and B together have 10 cards more than what D and E together have." \[ A + B = D + E + 10 \quad \text{(3)} \] - Statement 4: "B has two cards more than what C has." \[ B = C + 2 \quad \text{(4)} \] - Statement 5: "The total number of cards is 133." \[ A + B + C + D + E = 133 \quad \text{(5)} \] Step 3: Express All Variables in Terms of \( B \) - From (1): \( E = B - 3 \). - From (2): \( D = B + 3 \). - From (4): \( C = B - 2 \). - Substituting \( D \) and \( E \) into (3): \[ A + B = (B + 3) + (B - 3) + 10 \] Simplifying: \[ A + B = 2B + 10 \] Solving for \( A \): \[ A = B + 10 \] Step 4: Substitute All Variables into (5) Substituting \( A = B + 10 \), \( C = B - 2 \), \( D = B + 3 \), and \( E = B - 3 \) into equation (5): \[ (B + 10) + B + (B - 2) + (B + 3) + (B - 3) = 133 \] Simplifying: \[ 5B + 8 = 133 \] Solving for \( B \): \[ 5B = 125 \] \[ B = 25 \] Final Answer: The number of cards B has is \(\boxed{25}\).