W and T together can do a work in 140 days. W alone can do the work in 160 days. In how many days can T alone do the work?

Options

A. 1000
B. 1200
C. 1120
D. 1500

Answer

To find out how many days T alone can do the work, we can follow these steps:

### Step 1: Determine the work done by W and T together
W and T together can complete the work in 140 days. Therefore, their combined work rate is:
\[ \text{Work rate of W and T} = \frac{1}{140} \text{ (work per day)} \]

### Step 2: Determine the work done by W alone
W alone can complete the work in 160 days. Therefore, W's work rate is:
\[ \text{Work rate of W} = \frac{1}{160} \text{ (work per day)} \]

### Step 3: Calculate the work done by T alone
Let T's work rate be \( \frac{1}{x} \) (where \( x \) is the number of days T takes to complete the work alone). The combined work rate of W and T can be expressed as:
\[ \text{Work rate of W} + \text{Work rate of T} = \text{Work rate of W and T} \]
Substituting the values we have:
\[ \frac{1}{160} + \frac{1}{x} = \frac{1}{140} \]

### Step 4: Solve for \( x \)
To solve for \( x \), we need to find a common denominator and rearrange the equation:
\[ \frac{1}{x} = \frac{1}{140} - \frac{1}{160} \]
Finding a common denominator for 140 and 160, which is 1120:
\[ \frac{1}{140} = \frac{8}{1120}, \quad \frac{1}{160} = \frac{7}{1120} \]
Now substituting these values:
\[ \frac{1}{x} = \frac{8}{1120} - \frac{7}{1120} = \frac{1}{1120} \]

### Step 5: Calculate \( x \)
Taking the reciprocal gives us:
\[ x = 1120 \]
Thus, T alone can complete the work in 1120 days.

### Final Answer:
T alone can do the work in **1120 days**.

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