Question

Pintoo dealt some cards to Minto and himself from a full pack of playing cards and laid the rest aside. Pintoo then said to Mintoo, "If you give me a certain number of your cards I will have 4 times as many cards as you have. If I give you the same number of cards, I will have thrice as many cards as you have." How many cards did Pintoo have?

• A. 31
• B. 32
• C. 29
• D. 30

Answer 1

The Correct Option is A

The correct option is (A): 31
Explanation: Let's solve this step by step.
Let the number of cards Pintoo has be \( P \) and the number of cards Mintoo has be \( M \).
First condition:
If Mintoo gives a certain number of cards, say \( x \), to Pintoo, then Pintoo will have 4 times as many cards as Mintoo. After the exchange:
- Pintoo will have \( P + x \) cards
- Mintoo will have \( M - x \) cards
According to the condition:
\[P + x = 4(M - x)\]
Expanding the equation:
\[P + x = 4M - 4x\]
Simplifying:
\[P + 5x = 4M \quad \text{(Equation 1)}\]
Second condition:
If Pintoo gives \( x \) cards to Mintoo, then Pintoo will have 3 times as many cards as Mintoo. After the exchange:
- Pintoo will have \( P - x \) cards
- Mintoo will have \( M + x \) cards
According to the second condition:
\[P - x = 3(M + x)\]
Expanding this equation:
\[P - x = 3M + 3x\]
Simplifying:
\[P - 4x = 3M \quad \text{(Equation 2)}\]
Solving the system of equations:
From Equation 1:
\[P + 5x = 4M \quad \text{(Equation 1)}\]
From Equation 2:
\[P - 4x = 3M \quad \text{(Equation 2)}\]
Subtract Equation 2 from Equation 1:
\[(P + 5x) - (P - 4x) = 4M - 3M\]
\[9x = M\]
Now substitute \( M = 9x \) into Equation 1:
\[P + 5x = 4(9x)\]
\[P + 5x = 36x\]
\[P = 31x\]
Since \( x = 1 \) (the smallest number of cards exchanged), Pintoo has:
\[P = 31\]
Conclusion:
Pintoo had 31 cards, so the correct answer is Option A: 31.