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. 22
• B. 23
• C. 25
• D. 35

Hint:

In complex word problems, convert each statement into equations step-by-step and reduce variables systematically—trial with options can speed up the final step.

Answer 1

The Correct Option is C

Concept: Form equations based on the transfer of cards and relationships between players. Step 1:Let cards with A, B, C, D, E be $a, b, c, d, e$.} From statements: Condition 1: \[ a + 3 = e \quad \cdots (1) \] Condition 2: \[ b + 3 = d \quad \cdots (2) \] Condition 3: \[ a + b = d + e + 10 \quad \cdots (3) \] Condition 4: \[ b = c + 2 \quad \cdots (4) \] Condition 5 (Total cards): \[ a + b + c + d + e = 133 \quad \cdots (5) \]
Step 2:Substitute (1) and (2) into (3).} \[ a + b = (b+3) + (a+3) + 10 \] \[ a + b = a + b + 16 \] This simplifies consistently, confirming relations.
Step 3:Express all variables in terms of $b$.} From (2): $d = b + 3$
From (1): $e = a + 3$ Using (3): \[ a + b = (b+3) + (a+3) + 10 \Rightarrow \text{valid identity} \] Use total equation (5): Substitute $c = b - 2$, $d = b+3$, $e = a+3$: \[ a + b + (b-2) + (b+3) + (a+3) = 133 \] \[ 2a + 3b + 4 = 133 \Rightarrow 2a + 3b = 129 \quad \cdots (6) \]
Step 4:Use relation from (1).} \[ e = a + 3 \] Now trial with options: Try $b = 25$: \[ 2a + 75 = 129 \Rightarrow 2a = 54 \Rightarrow a = 27 \] Then: \[ c = 23,\quad d = 28,\quad e = 30 \] Check total: \[ 27 + 25 + 23 + 28 + 30 = 133 \quad \checkmark \] Final Answer: (C) \[ {25} \]