A can do a piece of work in 90 days, B in 40 days and C in 12 days. They work for a day each in turn i.e., first day A does it alone, B does it the second day and C the third day. After that A does it for another day, and so on. After finishing the work they get Rs. 240. If the wages are divided in proportion to the work done by them, find what each will get ?

Solution(By Examveda Team)

$$\eqalign{ & \left( {{\text{A}} + {\text{B}} + {\text{C}}} \right){\text{'s 3 days work}} \cr & = \frac{1}{{90}} + \frac{1}{{40}} + \frac{1}{{12}} \cr & = \frac{{43}}{{360}} \cr & \left( {{\text{A}} + {\text{B}} + {\text{C}}} \right){\text{'s 24 days work}} \cr & = \frac{{43}}{{360}} \times 8 \cr & = \frac{{344}}{{360}} \cr & {\text{Remaining work}} \cr & = \left( {1 - \frac{{344}}{{360}}} \right) \cr & = \frac{{16}}{{360}} \cr & = \frac{4}{{90}} \cr & {\text{On 25th day, it is A's turn}}{\text{.}} \cr & {\text{A's 1 day's work}} = \frac{1}{{90}} \cr & {\text{Remaining work}} \cr & = \left( {\frac{4}{{90}} - \frac{1}{{90}}} \right) \cr & = \frac{3}{{90}} \cr & = \frac{1}{{30}} \cr & {\text{On 26th day, it is B's turn}}{\text{.}} \cr & {\text{B's 1 day's work}} = \frac{1}{{40}} \cr & {\text{Remaining work}} \cr & = \left( {\frac{1}{{30}} - \frac{1}{{40}}} \right) \cr & = \frac{1}{{120}} \cr & {\text{On 27th day, it is C's turn}}{\text{.}} \cr & \frac{1}{{12}}{\text{ work is done by C in 1 day}}{\text{.}} \cr & \frac{1}{{120}}{\text{ work is done by C in }} \cr & = \left( {12 \times \frac{1}{{120}}} \right) \cr & = \frac{1}{{10}}{\text{ day}}{\text{.}} \cr} $$
Hence, the whole work is complete in $${\text{26}}\frac{1}{{10}}$$ days out of which A worked for 9 days,
B worked for 9 days and C worked for $${\text{8}}\frac{1}{{10}}$$ days.
Ratio of wages of A, B and C = Ratio of work done by A, B and C
$$ = \left( {\frac{1}{{90}} \times 9} \right):\left( {\frac{1}{{40}} \times 9} \right)$$     : $$\left( {\frac{1}{{12}} \times 8\frac{1}{{10}}} \right)$$
$$\eqalign{ & = \frac{1}{{10}}:\frac{9}{{40}}:\frac{{27}}{{40}} \cr & = 4:9:27 \cr & \therefore {\text{A's share}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{4}{{40}} \times 240} \right) \cr & = {\text{Rs}}{\text{.24}} \cr & {\text{B's share}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{9}{{40}} \times 240} \right) \cr & = {\text{Rs}}.54 \cr & {\text{C's share}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{27}}{{40}} \times 240} \right) \cr & = {\text{Rs}}{\text{.162}} \cr} $$