In a partnership business, A invests ${\frac{1}{6}}^{\text{th}}$ of the capital for $\frac{1}{6}$ of the total time, B invests ${\frac{1}{4}}^{\text{th}}$ of the capital for $\frac{1}{4}$ of the total time and C, the rest of the capital for the whole time. Find the share of B, out of a profit of Rs. 1940.

Let the total capital be $K$ and the total time be $T$.
According to the question:

* Investment of A ($I_A$) = $\frac{1}{6}K$, Time for A ($t_A$) = $\frac{1}{6}T$
* Investment of B ($I_B$) = $\frac{1}{4}K$, Time for B ($t_B$) = $\frac{1}{4}T$
* Investment of C ($I_C$) = Remaining capital = $K-\left(\frac{1}{6}K+\frac{1}{4}K\right)=K-\frac{5}{12}K=\frac{7}{12}K$
* Time for C ($t_C$) = Whole time = $T$

The profit is divided in the ratio of the product of capital and time:

$$\text{Profit Ratio (A : B : C)}=(I_A\times t_A):(I_B\times t_B):(I_C\times t_C)$$

$$\text{Ratio}=\left(\frac{1}{6}K\times \frac{1}{6}T\right):\left(\frac{1}{4}K\times \frac{1}{4}T\right):\left(\frac{7}{12}K\times T\right)$$

$$\text{Ratio}=\frac{1}{36}:\frac{1}{16}:\frac{7}{12}$$

To simplify the ratio, multiply by the LCM of 36, 16, and 12, which is 144:

$$\text{Ratio}=\left(\frac{1}{36}\times 144\right):\left(\frac{1}{16}\times 144\right):\left(\frac{7}{12}\times 144\right)$$

$$\text{Ratio}=4:9:84$$

Total units of profit = $4+9+84=97$ units

Given total profit = Rs. 1940

$$\text{Value of 1 unit}=\frac{1940}{97}=\text{Rs. }20$$

Therefore, the share of B = $9\times 20=\text{Rs. }180$