Let the total work be denoted as $W$, and the work done by X and Y together in one day is their combined efficiency, which is $\frac{W}{12}$.

Step 1: Work done by X and Y together in 4 days

In 4 days, X and Y together will complete: $\text{Work completed in 4 days} = 4 \times \frac{W}{12} = \frac{W}{3}$ This means that $\frac{2W}{3}$ of the work remains after Y leaves the job.

Step 2: Work done by X and Z in the remaining 8 days

Z joins the work, and the efficiency of Z is $\frac{5}{3}$ times the efficiency of X. Let the efficiency of X be $E_X$. Then the efficiency of Z is: $E_Z = \frac{5}{3} E_X$ The combined efficiency of X and Z is: $E_X + E_Z = E_X + \frac{5}{3} E_X = \frac{8}{3} E_X$ This combined efficiency of X and Z completes $\frac{8}{3} E_X$ of the work per day. Over 8 days, the work they complete is: $\text{Work completed by X and Z in 8 days} = 8 \times \frac{8}{3} E_X = \frac{64}{3} E_X$ But we know that the remaining work after Y leaves is $\frac{2W}{3}$. Therefore: $\frac{64}{3} E_X = \frac{2W}{3}$ Solving for $E_X$: $E_X = \frac{2W}{64} = \frac{W}{32}$

Step 3: Efficiency of Y

From the total work equation, we know that X and Y together complete $\frac{W}{12}$ of the work per day. Thus: $E_X + E_Y = \frac{W}{12}$ Substitute $E_X = \frac{W}{32}$: $\frac{W}{32} + E_Y = \frac{W}{12}$ Solve for $E_Y$: $E_Y = \frac{W}{12} - \frac{W}{32} = \frac{8W - 3W}{96} = \frac{5W}{96}$

Step 4: Time taken by Y alone to complete the work

If Y works alone, the time taken to complete the work is: $\text{Time taken by Y} = \frac{W}{E_Y} = \frac{W}{\frac{5W}{96}} = \frac{96}{5} = 19.2 \text{ days}$

Thus, Y can complete the whole work alone in 19.2 days.

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Would you like any further details on the steps? Here are some related questions:

1. How do you calculate the total work in such problems?
2. How does the change in efficiency of workers affect the time to complete work?
3. What if the efficiencies of all workers were proportional to another unit of work?
4. How can you solve for multiple unknown efficiencies simultaneously?
5. Can such problems be generalized to more than 3 workers?

Tip: Always double-check units when calculating efficiencies to avoid errors in the final steps!