X and Y can complete a work in 5 days. Y and Z can complete the same work in 6 days. Z and X can complete the same work in `(15)/(2)` days. In how many days will X, Y and Z together complete the work?

Options

A. A)2
B. B)4
C. C)3
D. D)`(2)/(5)`

Answer

To solve the problem, we need to find out how many days X, Y, and Z together will complete the work based on the information given about their individual and combined efficiencies.

### Step-by-Step Solution:

1. **Understanding the Work Rates**:
- Let the total work be represented as 1 unit of work.
- Let the work done by X in one day be \( x \), by Y be \( y \), and by Z be \( z \).

2. **Setting Up Equations**:
- From the problem, we have the following equations based on the combined work rates:
- \( x + y = \frac{1}{5} \) (since X and Y can complete the work in 5 days)
- \( y + z = \frac{1}{6} \) (since Y and Z can complete the work in 6 days)
- \( z + x = \frac{2}{15} \) (since Z and X can complete the work in \( \frac{15}{2} \) days)

3. **Solving the Equations**:
- We can rewrite the equations:
- \( x + y = \frac{1}{5} \) (1)
- \( y + z = \frac{1}{6} \) (2)
- \( z + x = \frac{2}{15} \) (3)

4. **Finding Individual Work Rates**:
- From equation (1), we can express \( y \) in terms of \( x \):
\[ y = \frac{1}{5} - x \]
- Substitute \( y \) into equation (2):
\[ \left(\frac{1}{5} - x\right) + z = \frac{1}{6} \]
Simplifying gives:
\[ z = \frac{1}{6} - \frac{1}{5} + x = x - \frac{1}{30} \]
- Substitute \( z \) into equation (3):
\[ \left(x - \frac{1}{30}\right) + x = \frac{2}{15} \]
Combining like terms gives:
\[ 2x - \frac{1}{30} = \frac{2}{15} \]
Converting \( \frac{2}{15} \) to a fraction with a denominator of 30:
\[ 2x - \frac{1}{30} = \frac{4}{30} \]
Thus:
\[ 2x = \frac{4}{30} + \frac{1}{30} = \frac{5}{30} = \frac{1}{6} \]
Therefore:
\[ x = \frac{1}{12} \]

5. **Finding \( y \) and \( z \)**:
- Substitute \( x \) back to find \( y \):
\[ y = \frac{1}{5} - \frac{1}{12} = \frac{12 - 5}{60} = \frac{7}{60} \]
- Substitute \( y \) to find \( z \):
\[ z = \frac{1}{6} - y = \frac{1}{6} - \frac{7}{60} = \frac{10 - 7}{60} = \frac{3}{60} = \frac{1}{20} \]

6. **Total Work Rate of X, Y, and Z**:
- Now, we can find the combined work rate:
\[ x + y + z = \frac{1}{12} + \frac{7}{60} + \frac{1}{20} \]
Finding a common denominator (60):
\[ = \frac{5}{60} + \frac{7}{60} + \frac{3}{60} = \frac{15}{60} = \frac{1}{4} \]

7. **Calculating Days to Complete Work**:
- The time taken by X, Y, and Z together to complete 1 unit of work:
\[ \text{Days} = \frac{1 \text{ unit}}{\frac{1}{4}} = 4 \text{ days} \]

### Final Answer:
X, Y, and Z together will complete the work in **4 days**.