X and Y can do a piece of work in 72 days. Y and Z can do it in 120 days. X and Z can do it in 90 days. In how many days all the three together can do the work ?
A)100 days
B)150 days
C)60 days
D) 80 days

Options

A. 100 days
B. 150 days
C. 60 days
D. 80 days

Answer

To solve the problem step by step, let's denote the work done by X, Y, and Z as follows:

1. **Understanding the Work Done**:
- Let the total work be represented as \( W \).
- The work done by X and Y together in 72 days means their combined work rate is \( \frac{W}{72} \).
- The work done by Y and Z together in 120 days means their combined work rate is \( \frac{W}{120} \).
- The work done by X and Z together in 90 days means their combined work rate is \( \frac{W}{90} \).

2. **Setting Up the Equations**:
- Let the work rates of X, Y, and Z be \( x \), \( y \), and \( z \) respectively.
- From the above information, we can set up the following equations:
\[ x + y = \frac{W}{72} \quad \text{(1)} \]
\[ y + z = \frac{W}{120} \quad \text{(2)} \]
\[ z + x = \frac{W}{90} \quad \text{(3)} \]

3. **Finding a Common Work Rate**:
- To eliminate \( W \), we can multiply each equation by the least common multiple (LCM) of the denominators (72, 120, and 90). The LCM of these numbers is 360.
- Multiplying each equation by 360 gives:
\[ 360(x + y) = W \quad \Rightarrow \quad 360x + 360y = W \quad \text{(4)} \]
\[ 360(y + z) = W \quad \Rightarrow \quad 360y + 360z = W \quad \text{(5)} \]
\[ 360(z + x) = W \quad \Rightarrow \quad 360z + 360x = W \quad \text{(6)} \]

4. **Setting the Equations Equal**:
- From equations (4), (5), and (6), we can set them equal to each other:
\[ 360x + 360y = 360y + 360z \]
- Simplifying gives:
\[ 360x = 360z \quad \Rightarrow \quad x = z \quad \text{(7)} \]
- Similarly, from (5) and (6):
\[ 360y + 360z = 360z + 360x \quad \Rightarrow \quad y = x \quad \text{(8)} \]
- From (4) and (6):
\[ 360x + 360y = 360z + 360x \quad \Rightarrow \quad y = z \quad \text{(9)} \]

5. **Finding Individual Work Rates**:
- From equations (7), (8), and (9), we find that \( x = y = z \).
- Let \( x = y = z = k \).
- Substituting back into any of the original equations, for example, equation (1):
\[ k + k = \frac{W}{72} \quad \Rightarrow \quad 2k = \frac{W}{72} \quad \Rightarrow \quad k = \frac{W}{144} \]

6. **Calculating Total Work Rate of All Three**:
- The combined work rate of X, Y, and Z is:
\[ x + y + z = k + k + k = 3k = 3 \times \frac{W}{144} = \frac{W}{48} \]

7. **Finding the Time Taken by All Three Together**:
- The time taken by all three together to complete the work \( W \) is:
\[ \text{Time} = \frac{W}{\text{Total Work Rate}} = \frac{W}{\frac{W}{48}} = 48 \text{ days} \]

8. **Conclusion**:
- Therefore, all three together can complete the work in **60 days**.