X and Y together can complete a work in 60 days. They started the work together but after 30 days X left the work. If the remaining work is completed by Y in 75 more days, then X alone can complete the entire work in how many days?

Correct option is C

Given:

• X and Y together can complete the work in 60 days.

• They worked together for 30 days.

• Y completed the remaining work in 75 days.

Concept Used:

Let the total work be (W), and the efficiencies of X and Y be $$(E_X$$​ and$$E_Y$$​  respectively.

Work done by any person is calculated as:

Work = $$Efficiency \times Time$$​​

Given that $$(E_X + E_Y )= \frac{1}{60}$$​, since they can together complete the work in 60 days.

Solution:

Work done by X and Y together in 30 days:

​$$Work_{\text{together}} = (E_X + E_Y) \times 30 = \frac{30}{60} = \frac{1}{2}$$​​

Thus, half of the work is completed in 30 days.

Remaining work:

Remaining Work = $$1 - \frac{1}{2} = \frac{1}{2}$$​​

Work completed by Y in 75 days:

​$$E_Y \times 75 = \frac{1}{2}$$​​

Thus,  $$E_Y = \frac{\frac{1}{2}}{75} = \frac{1}{150}.$$​​

Efficiency of X:

From  $$E_X + E_Y = \frac{1}{60},$$​​

​$$E_X = \frac{1}{60} - E_Y = \frac{1}{60} - \frac{1}{150}$$​​

Taking LCM: $$E_X = \frac{5 - 2}{300} = \frac{3}{300} = \frac{1}{100}$$​​

Time taken by X alone to complete the work:

Time =  $$\frac{\text{Work}}{\text{Efficiency}} = \frac{1}{E_X} = \frac{1}{\frac{1}{100}} = 100$$​​

X alone can complete the entire work in 100 days.