Question

Mohan takes 16 days less than Manoj to do a piece of work. If both working together can do it in 15 days, in how many days will Mohan alone complete the work?
A. 24 days.
B. 22 days.
C. 20 days.
D. 26 days.

Answer 1

Hint:
In this question, we need to form equations from the given conditions and then solve them by relevant methods. This question can be solved by using two very important methods of solving linear and quadratic equations. First, we will apply the unitary method to find the amount of work each Mohan and Manoj do in one day. Secondly, we apply the factorization method to solve the quadratic equation so formed.

Complete step by step solution:
Let the number of days taken by Mohan to do the piece of work\[ = x\]
Then according to question, number of days taken by Manoj to do the work\[ = x + 16\]
If we consider both of them work at a uniform pace then,
Part of work done by Mohan in one day\[ = \dfrac{1}{x}\] (here we used the unitary method to find the work done in 1 day)
Similarly, part of work done by Manoj in one day\[ = \dfrac{1}{{x + 16}}\].
Now, we are given that both of them can complete the work in 15 days if they work together.
Therefore, the part of work done by both of them together =\[\dfrac{1}{{15}}\].
As a result,
\[
\dfrac{1}{x} + \dfrac{1}{{x + 16}} = \dfrac{1}{{15}}\\
\Rightarrow \dfrac{{x + 16 + x}}{{x\left( {x + 16} \right)}} = \dfrac{1}{{15}}\\
\Rightarrow \dfrac{{2x + 16}}{{{x^2} + 16x}} = \dfrac{1}{{15}}\\
\Rightarrow 30x + 240 = {x^2} + 16x\\
\Rightarrow {x^2} - 14x - 240 = 0
\]
(Here we have a quadratic equation which can be solved by factorization)
\[
\Rightarrow {x^2} - 24x + 10x - 240 = 0\\
\Rightarrow x\left( {x - 24} \right) + 10\left( {x - 24} \right) = 0\\
\Rightarrow \left( {x - 24} \right)\left( {x + 10} \right) = 0
\]
Therefore, \[x = 24\] or \[x = - 10\], but here ‘x’ denotes the number of days which cannot be negative.
So \[x = 24\], i.e. Mohan will take 24 days to work alone.

The correct option is A. 24 days

Note:
In this question, in the initial steps, we can also consider ‘x’ to be the number of days taken by Manoj to do work. In that case, the number of days taken by Mohan will be ‘x-16,’ and then we solve further as we did above. We will arrive at the same result in both cases. Also, we can solve the quadratic equation so formed by the discriminant method instead of the factorization method.