In a partnership, X invests \(\frac{1}{6}\)^th of the capital for \(\frac{1}{6}\)^th of the time, Y invests \(\frac{1}{3}\)rd of the capital for \(\frac{1}{3}\)^rd time and Z invests the remaining capital for the whole time. If at the end of the year the profit earned is ₹ 23,000, then what will be Y's share ?

Given:

X invests (1/6)th of the capital for (1/6)th of the time

Y invests (1/3)rd of the capital for (1/3)rd of the time

Z invests the remaining capital for the whole time

Total profit at the end of the year = ₹23,000

Formula used:

Profit share is based on the product of capital invested and time

Calculation:

X's share: (1/6) × (1/6) = 1/36

Y's share: (1/3) × (1/3) = 1/9

Z's share: Remaining capital = 1 - (1/6 + 1/3) = 1 - 1/2 = 1/2

Z's time = 1 year

Z's share: (1/2) × 1 = 1/2

Total shares: 1/36 + 1/9 + 1/2

⇒ Total shares = 1/36 + 4/36 + 18/36 = 23/36

Y's share fraction = (1/9) / (23/36)

⇒ Y's share = (1/9) × (36/23) = 4/23

Total profit = ₹23,000

Y's share in profit = (4/23) × 23,000

⇒ Y's share = ₹4,000

∴ The correct answer is option (1).