### Theory:

A group of people work together, and they get a share of money for the work they do is called as sharing the money for the work.
If $$A$$ and $$B$$ do work in $$x$$ and $$y$$ days, respectively, then the ratio of the work done by $$A$$ and $$B$$ is equal to the ratio of their wages. That is, $$\frac{1}{x} : \frac{1}{y} = y : x$$
If $$A$$, $$B$$ and $$C$$ do the same work in $$x$$, $$y$$ and $$z$$ days, respectively, their share of money is calculated in the ratio $$\frac{1}{x}:\frac{1}{y}:\frac{1}{z}$$.
Example:
If $$A$$ can finish a work in $$10$$ days, $$B$$ can finish a work in $$15$$ days, and $$C$$ can finish a work in $$20$$ days. All three worked together and earned a salary of $$₹6500$$. Find the amount of money earned by $$A$$, $$B$$ and $$C$$.

Solution:

Given:

$$A$$ can finish a work in $$10$$ days.

$$B$$ can finish a work in $$15$$ days.

$$C$$ can finish a work in $$20$$ days.
The ratio of the work done by $$A$$, $$B$$ and $$C$$ is equal to the ratio of their wages.
Thus, we have:

$$\frac{1}{10} : \frac{1}{15} : \frac{1}{20} =$$ $$\frac{6}{60} : \frac{4}{60} : \frac{3}{60}$$

$$= 6 : 4 : 3$$

The total parts $$= 6 + 4 + 3 = 13$$

$$A$$'s share $$= \frac{6}{13} \times 6500 = ₹3000$$

$$B$$'s share $$= \frac{4}{13} \times 6500 = ₹2000$$

$$C$$'s share $$= \frac{3}{13} \times 6500 = ₹1500$$

Therefore, the shares of $$A$$, $$B$$ and $$C$$ are $$₹3000$$, $$₹2000$$ and $$₹1500$$ respectively.