A can do a work in 12 hours, B can do it in 14 hours. They work alternatively starting from A. A earns Rs. $15/hr.$ How much does he earn from the whole job?

Hint: To solve this question first we will calculate the total work by taking the LCM of time taken to complete the work by A and B. Then we will calculate individually the total work done by A and B. Then by multiplying the total work by A’s hourly earning we will get the desired answer.

Complete step by step answer:
We have been given that A can do a work in 12 hours, B can do in 14 hours. They work alternatively starting from A. A earns Rs. $15/hr.$
We have to find the total earning A gets from the job.
Now, let us first calculate the total work by taking the LCM of time taken to complete the work by A and B. then we will get
$\Rightarrow \text{Total work}=\text{LCM of }\left( 12,14 \right)$
Now, the LCM of 12 and 14 will be
$\begin{align}
  & 2\left| \!{\underline {\,
  12,14 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6,7 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3,7 \,}} \right. \\
 & 7\left| \!{\underline {\,
  1,7 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1,1 \,}} \right. \\
\end{align}$
LCM of 12 and 14 will be 84.
So the total work will be 84 units.
Now, the work done by A will be
$\begin{align}
  & \Rightarrow \dfrac{84}{12} \\
 & \Rightarrow 7units \\
\end{align}$
Now, the work done by B will be
$\begin{align}
  & \Rightarrow \dfrac{84}{14} \\
 & \Rightarrow 6 units \\
\end{align}$
Now, given in the question they work alternatively starting from A. So on the first day A completes 7 units and the next day B completes 6 units. It means in 2 days A and B complete $7+6=13units$.
Let us consider A worked 6 days and B worked 6 days so in 12 days the total work done by them will be
$\begin{align}
  & \Rightarrow 13\times 6 \\
 & \Rightarrow 78 units \\
\end{align}$
Now, the next day i.e. on the seventh day it’s A’s turn and he completed the remaining work i.e. $6units$ in $\dfrac{6}{7}hours$.
Now, A earns Rs. $15/hr.$ So the total amount he earns from the job will be
$\begin{align}
  & \Rightarrow 12\times 15\times 3+15\times \dfrac{6}{7} \\
 & \Rightarrow 540+13approx \\
 & \Rightarrow 553 \\
\end{align}$
Hence A gets Rs. 553 approx from the whole job.

Note: The key concept to solve this type of question is to first calculate the total work if not given in the question. Then calculate the work done by a person in one day. Then solve further according to the demand of the question.