Given:
Four persons A, B, C, and D play a game of cards.
We need to determine the number of cards A has.
Calculation:
Let:
A = Number of cards with A
B = Number of cards with B
C = Number of cards with C
D = Number of cards with D
Forming the equations based on given conditions:
- If A gives 8 cards to B, then B will have as many as C:
⇒ B + 8 = C ...(i)
- After giving 8 cards, A will have 3 less than C:
⇒ A - 8 = C - 3 ...(ii)
- If A takes 6 cards from C, he will have twice as many as D:
⇒ A + 6 = 2D ...(iii)
- The total number of cards with B and D is 50:
⇒ B + D = 50 ...(iv)
Solving the equations:
From equation (i):
C = B + 8
Substituting this in equation (ii):
A - 8 = (B + 8) - 3
⇒ A - 8 = B + 5
⇒ A = B + 13 ...(v)
Substituting equation (i) in equation (iii):
A + 6 = 2D
Using equation (v):
(B + 13) + 6 = 2D
⇒ B + 19 = 2D
⇒ D = (B + 19) / 2 ...(vi)
Substituting equation (vi) in equation (iv):
B + (B + 19) / 2 = 50
Multiplying by 2 to eliminate fraction:
2B + B + 19 = 100
3B + 19 = 100
⇒ 3B = 81
⇒ B = 27
Using B = 27 in equation (i):
C = 27 + 8 = 35
Using B = 27 in equation (v):
A = 27 + 13 = 40
Using B = 27 in equation (vi):
D = (27 + 19) / 2 = 23
Final Answer:
A = 40
B = 27
C = 35
D = 23
∴ A has 40 cards.