Question

A player holds 13 cards of four suits, of which seven are black and six are red. There are twice as many diamonds as spades and twice as many hearts as diamonds. How many clubs does he hold?

• A. 4
• B. 5
• C. 6
• D. 7

Hint:

In card problems, always separate suits into black and red first, then apply given ratios—this simplifies equation formation.

Answer 1

The Correct Option is C

Concept: This problem is based on relationships between card suits in a standard deck, using proportional conditions and total count constraints. Step 1:Express all suits in terms of one variable.
Let spades = x.
Then using the given ratio relationships:

• Diamonds = 2x
• Hearts = 4x

Clubs remains as the unknown variable.
Step 2:Use total card constraint.
\[ x + 2x + 4x + \text{Clubs} = 13 \Rightarrow 7x + \text{Clubs} = 13 \quad (1) \]
Step 3:Use black card condition.
Black cards = Spades + Clubs: \[ x + \text{Clubs} = 7 \quad (2) \]
Step 4:Eliminate variable and solve.
From (2): \[ \text{Clubs} = 7 - x \] Substitute into (1): \[ 7x + (7 - x) = 13 \Rightarrow 6x + 7 = 13 \Rightarrow 6x = 6 \Rightarrow x = 1 \] Now: \[ \text{Clubs} = 7 - 1 = 6 \]
Final Answer: 6