Question

Iqbal dealt some cards to Mushtaq and himself from a full pack of playing cards and laid the rest aside. Iqbal then said to Mushtaq, “If you give me a certain number of your cards, I will have four times as many cards as you will have. If I give you the same number of cards, I will have thrice as many cards as you will have.” Of the given choices, which could represent the number of cards with Iqbal?

• A. 9
• B. 31
• C. 12
• D. 35

Hint:

Translate conditional transfers into simultaneous equations and solve systematically.

Answer 1

The Correct Option is B

Let Iqbal have $x$ cards, Mushtaq have $y$ cards, and $z$ be the number of cards transferred in both scenarios. Case 1: Mushtaq gives $z$ cards to Iqbal \[ x + z = 4(y - z) \quad \Rightarrow \quad x + 5z = 4y \tag{1} \] Case 2: Iqbal gives $z$ cards to Mushtaq \[ x - z = 3(y + z) \quad \Rightarrow \quad x - 4z = 3y \tag{2} \] From (1) and (2): \[ 4y - 5z - 4z = 3y \quad \Rightarrow \quad y = 9z \] Substitute into (1): \[ x + 5z = 4(9z) \quad \Rightarrow \quad x = 36z - 5z = 31z \] For $z = 1$: $x = 31,\ y = 9$ works perfectly. Verification: - Mushtaq gives 1 card: $(31+1) = 32$, $(9-1) = 8$, $32 = 4\times 8$ - Iqbal gives 1 card: $(31-1) = 30$, $(9+1) = 10$, $30 = 3\times 10$ \fbox{Final Answer: 31 cards with Iqbal} %Quick tip