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Mohan can fix bulbs on poles on a stretch of road in 8 hours and Bhanu takes 10 hours to do the same work. They decide to work together. How much time will they take to complete the light fittings?

Last updated date: 13th Jun 2024
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Hint: We will find the efficiency of Mohan and Bhanu by using the time they require to complete the work. Then we know that they have decided to work together. So their efficiencies to complete the work will add up. We will use the value of this added efficiency to find the time required by them to complete the work together.

Complete step-by-step solution
Let us denote the work of fixing bulbs on a stretch of road by W. Let us describe a formula to calculate the efficiency of a person for doing a certain quantity of work in the following manner,
$\text{Efficiency of a person = }\dfrac{\text{total work done}}{\text{time taken to do the total work}}$
 Now, we know that Mohan takes 8 hours to complete the W quantity of work. We will calculate the efficiency of Mohan using the formula described above, as follows
$\text{Efficiency of Mohan =}\dfrac{\text{W}}{8}$
Next, we know that Bhanu takes 10 hours to complete work W. So using the efficiency formula again, we will calculate the efficiency of Bhanu. We will get
$\text{Efficiency of Bhanu = }\dfrac{\text{W}}{10}$
Mohan and Bhanu have decided to work together. So, their efficiency will be added. Let us denote this added efficiency by E. So we have the following,
$\text{E = }\dfrac{\text{W}}{8}+\dfrac{\text{W}}{10}=\dfrac{5\text{W+4W}}{40}=\dfrac{9}{40}\text{W}$
Now, we have to find the time taken by both of them while working together to finish work W with efficiency E. Substituting these values in the formula for efficiency, we get
$\text{E=}\dfrac{\text{W}}{\text{time taken to do work W}}$
Rearranging the above equation and substituting the value of E, we get
\[\text{Time taken to do work W =}\dfrac{\text{W}}{\left( \dfrac{9}{40}\text{W} \right)}\]
Simplifying the above equation, we have
\[\text{Time taken to do work W =}\dfrac{40}{9}\text{ hrs}\]
Hence, the time required by both of them working together to complete work W is $\dfrac{40}{9}$ hours.

Note: It is important to understand the relationship between work done, the time required to do the work, and the efficiency. Describing the formula in clear words makes the relations between the work, efficiency, and time easy to understand. We should be careful while doing the calculations to avoid making minor errors.