X,Y and Z take 18 dyas to complete a piece of work. If X works. If X works alone he finishes the work in 36 days and if Y works alone he finishes it in 60 days. Long will it take Z to complete the work alone ?
A. 78 days
B. 90 days
C. 96 days
D. 14 days

Options

A. B
B. A
C. D
D. C

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the work done by X and Y in one day.
- X can complete the work in 36 days, so the work done by X in one day is:
\[ \text{Work done by X in one day} = \frac{1}{36} \]

- Y can complete the work in 60 days, so the work done by Y in one day is:
\[ \text{Work done by Y in one day} = \frac{1}{60} \]

### Step 2: Let Z's time to complete the work alone be \( z \) days.
- Therefore, the work done by Z in one day is:
\[ \text{Work done by Z in one day} = \frac{1}{z} \]

### Step 3: Combine the work done by X, Y, and Z together.
- Together, X, Y, and Z can complete the work in 18 days, so their combined work done in one day is:
\[ \text{Combined work in one day} = \frac{1}{18} \]

### Step 4: Set up the equation.
- The equation combining the work done by X, Y, and Z in one day is:
\[ \frac{1}{36} + \frac{1}{60} + \frac{1}{z} = \frac{1}{18} \]

### Step 5: Solve for \( \frac{1}{z} \).
- First, we need to find a common denominator for the fractions on the left side. The least common multiple (LCM) of 36, 60, and 18 is 180.
- Rewrite each fraction with a denominator of 180:
\[ \frac{1}{36} = \frac{5}{180}, \quad \frac{1}{60} = \frac{3}{180}, \quad \frac{1}{18} = \frac{10}{180} \]
- Substitute these values into the equation:
\[ \frac{5}{180} + \frac{3}{180} + \frac{1}{z} = \frac{10}{180} \]
- Combine the fractions:
\[ \frac{8}{180} + \frac{1}{z} = \frac{10}{180} \]

### Step 6: Isolate \( \frac{1}{z} \).
- Subtract \( \frac{8}{180} \) from both sides:
\[ \frac{1}{z} = \frac{10}{180} - \frac{8}{180} = \frac{2}{180} \]
- Simplify:
\[ \frac{1}{z} = \frac{1}{90} \]

### Step 7: Solve for \( z \).
- Taking the reciprocal gives:
\[ z = 90 \]

### Conclusion:
- Therefore, Z alone can complete the work in **90 days**.

### Final Answer:
**B. 90 days**