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

Answer: (A)

31

Answer 2

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.