X, Y and Z together can complete a work in 20 days. Y and Z can complete the same work in 30 days. In how many days X alone can complete the same work?

Options

A. A)60
B. B)45
C. C)40
D. D)50

Answer

To solve the problem step-by-step, we need to find out how many days X alone can complete the work, given the information about X, Y, and Z working together, and Y and Z working together.

### Step-by-step Solution:

1. **Understand the work rates:**
- Let the total work be represented as 1 unit of work.
- If X, Y, and Z together can complete the work in 20 days, their combined work rate is:
\[ \text{Work rate of (X + Y + Z)} = \frac{1 \text{ unit}}{20 \text{ days}} = \frac{1}{20} \text{ units/day} \]
- If Y and Z together can complete the work in 30 days, their combined work rate is:
\[ \text{Work rate of (Y + Z)} = \frac{1 \text{ unit}}{30 \text{ days}} = \frac{1}{30} \text{ units/day} \]

2. **Set up the equation for X's work rate:**
- Let the work rate of X be represented as \( R_X \).
- We know that:
\[ R_X + R_Y + R_Z = \frac{1}{20} \]
\[ R_Y + R_Z = \frac{1}{30} \]

3. **Find X's work rate:**
- To find \( R_X \), we can subtract the second equation from the first:
\[ R_X + (R_Y + R_Z) - (R_Y + R_Z) = \frac{1}{20} - \frac{1}{30} \]
- This simplifies to:
\[ R_X = \frac{1}{20} - \frac{1}{30} \]

4. **Calculate the difference:**
- To subtract the fractions, we need a common denominator. The least common multiple of 20 and 30 is 60.
- Convert the fractions:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
- Now subtract:
\[ R_X = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]

5. **Determine the time taken by X alone:**
- Since \( R_X = \frac{1}{60} \) units/day, it means X can complete the work alone in:
\[ \text{Time taken by X} = \frac{1 \text{ unit}}{R_X} = \frac{1}{\frac{1}{60}} = 60 \text{ days} \]

### Final Answer:
X alone can complete the work in **60 days**.

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