- Let the initial amount of wine in the container be x liters.
- In each operation, 15 liters of wine is removed and replaced by 15 liters of water.
- This is a classic replacement problem and follows the formula:
Final quantity of wine = x × (1 – r/x)ⁿ
where r is the quantity removed and replaced each time, and n is the number of repetitions.
- Here, r = 15, n = 3, and final ratio of wine to water = 343 : 169
- So, final wine = x × (1 – 15/x)³ = (343/512) × x
Set up the equation:
x × (1 – 15/x)³ = (343/512) × x
Cancel x from both sides:
(1 – 15/x)³ = 343/512
Take cube root on both sides:
1 – 15/x = ∛(343/512)
1 – 15/x = 7/8
Solve the equation:
1 – 7/8 = 15/x
1/8 = 15/x
x = 15 × 8 = 120
Answer: 120 liters
The initial amount of wine in the container was 120 liters.
Discussion & Comments