Question

Saurabh can finish a work in 18 hours, while Vinod can complete the same work in 24 hours. How long will it take them together to complete this work if Saurabh is called away 2 hours before the completion of work, while Vinod continues with the work?

Answer 1

Hint: According to given in the question we have to find the time to complete the given work by both of them Saurabh and Vinod when Saurabh can finish a work in 18 hours, while Vinod can complete the same work in 24 hours and work if Saurabh is called away 2 hours before the completion of work, while Vinod continues with the work. So, first of all we have to find the total work done by Saurabh in 1 hour which can be obtained by dividing 24 hours by 1 hour.
Now, same as we have to find the total work done by Vinod in 1 hour which can be obtained by dividing 18 hours by 1 hour.
Now, we have to let the time taken by them together to complete the work $ = t$hours.
So now, we have to find the efficiency of Saurabh which can be obtain by the multiplication of work done by Saurabh in 1 hour with the total time taken and same as we have to find the efficiency of Vinod which can be obtain by the multiplication of work done by Vinod in 1 hour with the total time taken.
Now, we have to add both of the total works done by them to find the time taken together to complete the given work which we let is t hours.

Complete step-by-step answer:
Step 1: First of all we have to find the total work done by Saurabh in 1 hour as explained in the solution hint.
Work done by Saurabh done in hour $ = \dfrac{w}{{18}}.............(1)$
Step 2: Same as the step 1 now, we have to find the total work done by Vinod in 1 hour as explained in the solution hint.
Work done by Vinod done in hour $ = \dfrac{w}{{24}}.............(2)$
Step 3: Suppose time taken by both of them to complete the work is t hours so, we have to find the total work done by Saurabh in the given time t hours as explained in the solution hint.
$ = \dfrac{w}{{18}} \times (t - 2)$
Step 4: Suppose time taken by both of them to complete the work is t hours, so, we have to find the total work done by Vinod in the given time t hours as explained in the solution hint.
$ = \dfrac{w}{{24}} \times t$
Step 5: Now, we have to add the total work done by both of them together to find the time.
Hence,
$ \Rightarrow \dfrac{w}{{24}} \times t + \dfrac{w}{{18}} \times (t - 2) = w$
Now, eliminating the term w from the expression obtained just above,
\[
   \Rightarrow \dfrac{1}{{24}} \times t + \dfrac{1}{{18}} \times (t - 2) = 1 \\
   \Rightarrow \dfrac{t}{{24}} + \dfrac{{t - 2}}{{18}} = 1
 \]
Now, to find the value of t we have to find the L.C.M of the expression as obtained just above,
$ \Rightarrow \dfrac{{18t + 24t - 48}}{{18 \times 24}} = 1$
Applying cross-multiplication,
$
   \Rightarrow 42t = 24 \times 18 + 48 \\
   \Rightarrow 42t = 480 \\
   \Rightarrow t = \dfrac{{480}}{{42}} \\
   \Rightarrow t = 11.5hours
 $
Final solution: Hence, we have obtained the required time to complete the work by both of them together is $11.5{\text{hours}}$ when Saurabh can finish a work in 18 hours, while Vinod can complete the same work in 24 hours and Saurabh is called away 2 hours before the completion of work, while Vinod continues with the work.

They both will take a total time of $11.5{\text{hours}}$ to finish the work.

Note: Step 1: First of all we have to let the total time done by both of them together id t hours. Now, we have to find the L.C.M of the time taken by Saurabh which is 18 hours and the time taken by Vinod which is 24 hours and let the total work done is 72units.
Hence,
Vinod has completed 4 units of work in 1 hour and Saurabh has completed 3 units of work in 1 hour.
Step 2: As mentioned in the question Saurabh was called before 2 hours of the completion of work hence,
Saurabh completed $3(t - 2)$ units of work and Vinod completed $4t$ units of work.
Hence, now we have to calculate the total units of work done by both of them together.
$
   \Rightarrow 4t + 3t - 6 = 72 \\
   \Rightarrow 7t = 78 \\
   \Rightarrow t = \dfrac{{78}}{7} \\
   \Rightarrow t = 11.5hours
 $
Hence, we have obtained the total time taken by both of them to complete the work.