In a partnership, X invests $\frac{1}{6}\text{th}$ of the capital for $\frac{1}{6}\text{th}$ of the time, Y invests $\frac{1}{3}\text{rd}$ of the capital for $\frac{1}{3}\text{rd}$ 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?

Let the total capital be $6$ units and total time be $6$ units (chosen based on the denominators $6$ and $3$).
Investments of each partner:

* X's capital = $\frac{1}{6}\times 6=1\text{ unit}$
* Y's capital = $\frac{1}{3}\times 6=2\text{ units}$
* Z's capital = $6-(1+2)=3\text{ units}$

Time of investment for each partner:

* X's time = $\frac{1}{6}\times 6=1\text{ unit}$
* Y's time = $\frac{1}{3}\times 6=2\text{ units}$
* Z's time = $6\text{ units}$ (whole time)

The profit sharing ratio ($\text{Capital}\times \text{Time}$):

$$\text{X : Y : Z}=(1\times 1):(2\times 2):(3\times 6)=1:4:18$$

Total profit ratio sum = $1+4+18=23\text{ units}$.
Given total profit = ₹23,000.

$$\text{Value of 1 unit}=\frac{23000}{23}=\text{₹}1,000$$

Y's share (which is $4$ units):

$$\text{Y’s Share}=4\times 1000=\text{₹}4,000$$