Question

To complete a job, Babu needs $6$ more days than Abu. If both of them do the job together it takes $4$ days to complete it. How many days does one need, if they do the job separately?

Answer 1

Hint:  Here we have to find how many days each of them take to complete the job separately. 

Now we have to assume that one person’s time to finish the job is a variable, say ‘x’ and then we have to calculate the other person’s days based on the above assumption.

Then, we form the equations based on the given data and solve it to get the final answer.

Complete step-by-step solution:

Let the number of days required for Abu to finish the job be $x$.

Then the number of days required for Babu to finish the job will be $x + 6$

So, Abu finishes $\dfrac{1}{x}$ part of the job and Babu finishes $\dfrac{1}{{x + 6}}$ part of the job in one day.

Given that they can together finish the job in $4$ days, we have

$ \Rightarrow \dfrac{1}{x} + \dfrac{1}{{x + 6}} = \dfrac{1}{4}$ 

Now taking L.C.M of the left handed denominator, we get  

$ \Rightarrow \dfrac{{\left( {x + 6} \right) + x}}{{x(x + 6)}} = \dfrac{1}{4}$

On simplify the LHS we get,

$ \Rightarrow \dfrac{{2x + 6}}{{{x^2} + 6x}} = \dfrac{1}{4}$

Now we are doing cross multiply in both the sides

$ \Rightarrow 4(2x + 6) = {x^2} + 6x$

We need to multiply the terms by open brackets 

$ \Rightarrow 8x + 24 = {x^2} + 6x$

Now taken all the terms in left hand sides and subtract the same degree terms 

$ \Rightarrow {x^2} - 2x - 24 = 0$.

Thus, we get the equation form.

Now we have to factorize the equation ${x^2} - 2x - 24 = 0$

Here we have to split the \[x\] terms and we get,

$ \Rightarrow {x^2} - 6x + 4x - 24 = 0$

Taking the term as common in first two term and  as common on second two term we get

$ \Rightarrow x\left( {x - 6} \right) + 4\left( {x - 6} \right) = 0$

Taking the common term we get

$ \Rightarrow \left( {x - 6} \right)\left( {x + 4} \right) = 0$

By equating each bracket to zero, 

We get the roots, 

$ \Rightarrow x = 6$ or $x = - 4$

Time cannot be in negative quantity

Here, x is the number of days required for Abu to finish the jobs, so it is not a negative number.

Thus, $x = - 4$ is inadmissible.

So $x = 6$

Hence, Abu requires $6$ days and Babu requires $x + 6 = 6 + 6 = 12$ days to finish the job alone.

Note: These types of questions are solved by taking work done by that person in one day $ = {\dfrac{1}{n}^{th}}$ part of the work.

Also, we can add or subtract quantities of only work done. 

We cannot add or subtract time which is taken to finish work. 

We need to avoid any type of calculation mistakes like $x - 6 = 6 - 6 = 0$

Hence we get zero days which is not possible.