X, Y and Z can complete a work in 15, 45 and 90 days respectively. In how many days X, Y and Z together can complete the same work?

Options

A. 12
B. 11
C. 9
D. 10

Answer

To find out how many days X, Y, and Z together can complete the work, we first need to determine their individual work rates and then combine them.

### Step-by-Step Solution:

1. **Determine Individual Work Rates**:
- X can complete the work in 15 days. Therefore, the work rate of X is:
\[ \text{Work rate of X} = \frac{1}{15} \text{ (work per day)} \]
- Y can complete the work in 45 days. Therefore, the work rate of Y is:
\[ \text{Work rate of Y} = \frac{1}{45} \text{ (work per day)} \]
- Z can complete the work in 90 days. Therefore, the work rate of Z is:
\[ \text{Work rate of Z} = \frac{1}{90} \text{ (work per day)} \]

2. **Combine Work Rates**:
- To find the combined work rate of X, Y, and Z, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{15} + \frac{1}{45} + \frac{1}{90} \]

3. **Find a Common Denominator**:
- The least common multiple (LCM) of 15, 45, and 90 is 90. We will convert each fraction to have a denominator of 90:
\[ \frac{1}{15} = \frac{6}{90}, \quad \frac{1}{45} = \frac{2}{90}, \quad \frac{1}{90} = \frac{1}{90} \]
- Now, we can add the fractions:
\[ \text{Combined work rate} = \frac{6}{90} + \frac{2}{90} + \frac{1}{90} = \frac{6 + 2 + 1}{90} = \frac{9}{90} = \frac{1}{10} \]

4. **Calculate Time Taken to Complete Work**:
- Since the combined work rate of X, Y, and Z is \(\frac{1}{10}\) (work per day), it means they can complete the entire work in:
\[ \text{Time taken} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{1}{10}} = 10 \text{ days} \]

### Final Answer:
X, Y, and Z together can complete the work in **10 days**.

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