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Let's solve the problem step by step.

### Step 1: Define Variables
Let:
- The number of days taken by **X** to complete the work = \( x \)
- The number of days taken by **Y** to complete the work = \( y \)
- The number of days taken by **Z** to complete the work = \( z \)

From the given conditions:
1. **Efficiency of Y is 60% more than X**  
   - Efficiency is inversely proportional to time.  
   - If efficiency of X is \( E \), then efficiency of Y is **1.6E**  
   - This means \( y = \frac{x}{1.6} = \frac{5x}{8} \)

2. **Efficiency of Z is 12.5% less than Y**  
   - 12.5% less means \( 87.5\% \) of Y’s efficiency, which is \( \frac{7}{8} \) of Y’s efficiency.  
   - Since efficiency is inverse to time:  
     \[
     z = \frac{y}{\frac{7}{8}} = \frac{8y}{7} = \frac{8}{7} \times \frac{5x}{8} = \frac{5x}{7}
     \]

3. **Difference in time taken by X and Y is 27 days**  
   \[
   x - y = 27
   \]
   Substituting \( y = \frac{5x}{8} \):

   \[
   x - \frac{5x}{8} = 27
   \]

   \[
   \frac{8x - 5x}{8} = 27
   \]

   \[
   \frac{3x}{8} = 27
   \]

   \[
   3x = 216
   \]

   \[
   x = 72
   \]

   Substituting \( x = 72 \) into \( y = \frac{5(72)}{8} \):

   \[
   y = 45
   \]

   Substituting \( y = 45 \) into \( z = \frac{5(72)}{7} \):

   \[
   z = 40
   \]

### Step 2: Work Done Per Day
- Work done by X per day = \( \frac{1}{72} \)
- Work done by Y per day = \( \frac{1}{45} \)
- Work done by Z per day = \( \frac{1}{40} \)

Total work done per day:

\[
\frac{1}{72} + \frac{1}{45} + \frac{1}{40}
\]

Finding LCM of 72, 45, and 40:  
\( \text{LCM} = 360 \)

Converting fractions:

\[
\frac{1}{72} = \frac{5}{360}, \quad \frac{1}{45} = \frac{8}{360}, \quad \frac{1}{40} = \frac{9}{360}
\]

Adding them:

\[
\frac{5 + 8 + 9}{360} = \frac{22}{360}
\]

\[
= \frac{1}{16.36}
\]

### Step 3: Total Time Taken by All
Total time:

\[
\approx 16 \text{ days}
\]

Thus, the correct answer is **(b) 16 days**.

---
Would you like more explanation on any step?

### Related Questions:
1. How do we calculate work efficiency ratios?
2. What is the significance of inverse proportionality in time and efficiency?
3. How do we find LCM when dealing with fractions?
4. Can we solve this problem using algebraic equations directly?
5. How would the answer change if efficiency conditions were different?

**Tip:** When dealing with efficiency problems, always convert efficiency to time ratios before solving!

Efficiency of Z is 12.5% less than Y, while efficiency of Y is 60% more than X. If the difference between the number of days taken by X and Y to complete a work is 27 days, then find the time taken by all the three together to complete the same work?

This problem explores the relationship between efficiency, time, and work. Learn how to calculate the combined time taken by workers X, Y, and Z based on their efficiencies. Suitable for students in Grades 7-9.

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