S does half as much work as T in `(1)/(7)` of the time taken by T. If together they take 21 days to complete a work, how may days shall S take to complete that work alone?

Options

A. 26
B. 27
C. 28
D. 32

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the relationship between S and T
According to the problem, S does half as much work as T in \( \frac{1}{7} \) of the time taken by T.

Let the time taken by T to complete the work alone be \( y \) days. Then, the time taken by S to complete the same work alone will be \( x \) days.

From the information given:
- In \( \frac{1}{7}y \) days, S does half the work that T does in \( y \) days.
- Therefore, in one day, S's work can be expressed as:
\[ \text{Work done by S in one day} = \frac{1}{2} \times \frac{1}{y} = \frac{1}{2y} \]

### Step 2: Express S's work in terms of T's work
Since S takes \( \frac{1}{7}y \) days to do half the work of T, we can express S's work in terms of T's work:
- The work done by T in one day is \( \frac{1}{y} \).
- Therefore, in \( \frac{1}{7}y \) days, T does:
\[ \text{Work done by T in } \frac{1}{7}y \text{ days} = \frac{1}{7} \times \frac{1}{y} = \frac{1}{7y} \]
- Since S does half of this work:
\[ \text{Work done by S in one day} = \frac{1}{2} \times \frac{1}{7y} = \frac{1}{14y} \]

### Step 3: Set up the equation for combined work
When S and T work together, they complete the work in 21 days. Therefore, their combined work rate can be expressed as:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{21} \]

### Step 4: Substitute S's work rate into the equation
From the previous steps, we know:
\[ \frac{1}{x} = \frac{1}{14y} \]
Substituting this into the combined work equation gives:
\[ \frac{1}{14y} + \frac{1}{y} = \frac{1}{21} \]

### Step 5: Simplify the equation
To simplify, find a common denominator for the left side:
\[ \frac{1 + 14}{14y} = \frac{15}{14y} \]
Thus, we have:
\[ \frac{15}{14y} = \frac{1}{21} \]

### Step 6: Cross-multiply to solve for y
Cross-multiplying gives:
\[ 15 \times 21 = 14y \]
\[ 315 = 14y \]
Now, solving for \( y \):
\[ y = \frac{315}{14} = 22.5 \]

### Step 7: Find S's time to complete the work alone
Now, we need to find \( x \):
Using the relationship \( \frac{1}{x} = \frac{1}{14y} \):
\[ \frac{1}{x} = \frac{1}{14 \times 22.5} \]
Calculating \( 14 \times 22.5 = 315 \):
\[ \frac{1}{x} = \frac{1}{315} \]
Thus:
\[ x = 315 \]

### Conclusion
S will take 315 days to complete the work alone.