Smith and Thomas can do a piece of work in 72 days, Thomas and Clark can do it in 120 days, Smith and Clark can do it in 90 days. In what time can Smith, Thomas and Clark do it, working together?

Options

A. 60 days
B. 66 days
C. 75 days
D. 80 days

Answer

To solve the problem, we can use the concept of work rates. Let's denote the amount of work done by Smith, Thomas, and Clark per day as S, T, and C respectively.

### Step 1: Determine the work rates of each pair
1. **Smith and Thomas** can complete the work in 72 days. Therefore, their combined work rate is:
\[ S + T = \frac{1}{72} \text{ (work per day)} \]

2. **Thomas and Clark** can complete the work in 120 days. Therefore, their combined work rate is:
\[ T + C = \frac{1}{120} \text{ (work per day)} \]

3. **Smith and Clark** can complete the work in 90 days. Therefore, their combined work rate is:
\[ S + C = \frac{1}{90} \text{ (work per day)} \]

### Step 2: Set up the equations
Now we have three equations based on the work rates:
1. \( S + T = \frac{1}{72} \) (1)
2. \( T + C = \frac{1}{120} \) (2)
3. \( S + C = \frac{1}{90} \) (3)

### Step 3: Solve the equations
We can add all three equations together:
\[ (S + T) + (T + C) + (S + C) = \frac{1}{72} + \frac{1}{120} + \frac{1}{90} \]
This simplifies to:
\[ 2S + 2T + 2C = \frac{1}{72} + \frac{1}{120} + \frac{1}{90} \]
Dividing the entire equation by 2 gives:
\[ S + T + C = \frac{1}{2} \left( \frac{1}{72} + \frac{1}{120} + \frac{1}{90} \right) \]

### Step 4: Calculate the right side
Now we need to calculate the sum on the right side:
1. Find a common denominator for 72, 120, and 90. The least common multiple (LCM) of these numbers is 360.
2. Convert each fraction:
- \( \frac{1}{72} = \frac{5}{360} \)
- \( \frac{1}{120} = \frac{3}{360} \)
- \( \frac{1}{90} = \frac{4}{360} \)

Adding these gives:
\[ \frac{5}{360} + \frac{3}{360} + \frac{4}{360} = \frac{12}{360} = \frac{1}{30} \]

### Step 5: Final work rate
Now we have:
\[ S + T + C = \frac{1}{2} \cdot \frac{1}{30} = \frac{1}{60} \]

### Step 6: Find the time taken by all three together
If \( S + T + C = \frac{1}{60} \), then the time taken by Smith, Thomas, and Clark working together to complete the work is:
\[ \text{Time} = \frac{1}{\text{Work rate}} = \frac{1}{\frac{1}{60}} = 60 \text{ days} \]

### Conclusion
Smith, Thomas, and Clark can complete the work together in **60 days**.

---