Calculating Combined Job Completion Time

The problem asks for the total time X and Y would take to complete a job together. We are given information about X's individual capacity and how X and Y shared the work.

Understanding Work Rates

First, let's determine the work rate of X. If X can finish the entire job in $141$ days, then in one day, X completes:

$ \text{X's Rate} = \frac{1}{141} \text{ of the job per day} $

Work Done by X Alone

X worked alone for $57$ days. The portion of the job completed by X during this time is:

$ \text{Work by X} = \text{X's Rate} \times \text{Days worked} = \frac{1}{141} \times 57 = \frac{57}{141} $

Simplifying the fraction:

$ \frac{57}{141} = \frac{3 \times 19}{3 \times 47} = \frac{19}{47} $

So, X completed $19/47$ of the job.

Remaining Work and Y's Rate

The remaining work that needed to be completed is:

$ \text{Remaining Work} = 1 - \text{Work by X} = 1 - \frac{19}{47} = \frac{47 - 19}{47} = \frac{28}{47} $

This remaining work ($28/47$) was completed by Y in $84$ days. Therefore, Y's work rate is:

$ \text{Y's Rate} = \frac{\text{Remaining Work}}{\text{Days worked by Y}} = \frac{28/47}{84} = \frac{28}{47 \times 84} $

Simplifying Y's rate:

$ \text{Y's Rate} = \frac{28}{47 \times 3 \times 28} = \frac{1}{47 \times 3} = \frac{1}{141} \text{ of the job per day} $

Combined Work Rate

Now, we find the combined rate at which X and Y work together:

$ \text{Combined Rate} = \text{X's Rate} + \text{Y's Rate} = \frac{1}{141} + \frac{1}{141} = \frac{2}{141} \text{ of the job per day} $

Total Time for Combined Effort

The total time required for both X and Y to complete the job together is the inverse of their combined rate:

$ \text{Time Together} = \frac{\text{Total Work}}{\text{Combined Rate}} = \frac{1}{2/141} = \frac{141}{2} \text{ days} $

Calculating the final value:

$ \frac{141}{2} = 70.5 \text{ days} $

Both X and Y together would take $70.5$ days to complete the entire job.