Ram and Shyam work on a job together for four days and complete 60% of it. Ram takes leave then and Shyam works for eight more days to complete the job. How long would Ram take to complete the entire job alone ?

The correct option is 10 days.

Step-by-step Derivation:

Let us denote the total work to be completed as 100%.

According to the question, Ram and Shyam work together for 4 days and complete 60% of the job.
We can find the combined 1-day work rate of Ram and Shyam by dividing the completed work by the number of days:

$$\text{Combined 1-day work rate}=\frac{60\%}{4}=15\%\text{ per day}$$

If we represent Ram's daily work rate as R and Shyam's daily work rate as S, we have:

$$R+S=15\%$$

After 4 days of working together, the remaining work is:

$$100\%-60\%=40\%$$

Ram then takes leave, and Shyam completes this remaining 40% of the work by working alone for 8 more days.
Therefore, Shyam's 1-day work rate (S) is:

$$S=\frac{40\%}{8}=5\%\text{ per day}$$

Now, we can substitute Shyam's rate (S) back into the combined rate equation to find Ram's daily work rate (R):

$$R+5\%=15\%$$

$$R=15\%-5\%=10\%\text{ per day}$$

This means Ram alone completes 10% of the total work in one day.
To find the total number of days Ram would take to complete the entire job (100% work) alone:

$$\text{Number of days for Ram alone}=\frac{100\%}{10\%\text{ per day}}=10\text{ days}$$

Thus, Ram would take 10 days to complete the entire job alone.