X can complete a job in 12 days. If X and Y work together, they can complete the job in 6 days. Y alone can complete the job in

Options

A. 10 days
B. 12 days
C. 15 days
D. 18 days

Answer

To solve the problem, we need to determine how long it takes for Y to complete the job alone, given the information about X and their combined work rate.

### Step-by-Step Solution:

1. **Determine the work rate of X:**
- If X can complete the job in 12 days, then the work done by X in one day is:
\[ \text{Work rate of X} = \frac{1}{12} \text{ of the job per day} \]

2. **Determine the combined work rate of X and Y:**
- If X and Y together can complete the job in 6 days, then their combined work rate is:
\[ \text{Combined work rate of X and Y} = \frac{1}{6} \text{ of the job per day} \]

3. **Set up the equation for Y's work rate:**
- Let the work rate of Y be \( \frac{1}{y} \) (where \( y \) is the number of days Y takes to complete the job alone).
- The equation for their combined work rate is:
\[ \text{Work rate of X} + \text{Work rate of Y} = \text{Combined work rate} \]
\[ \frac{1}{12} + \frac{1}{y} = \frac{1}{6} \]

4. **Solve for Y's work rate:**
- To solve for \( \frac{1}{y} \), first isolate it:
\[ \frac{1}{y} = \frac{1}{6} - \frac{1}{12} \]
- Find a common denominator (which is 12):
\[ \frac{1}{6} = \frac{2}{12} \]
\[ \frac{1}{y} = \frac{2}{12} - \frac{1}{12} = \frac{1}{12} \]

5. **Calculate y:**
- Taking the reciprocal gives:
\[ y = 12 \text{ days} \]

### Final Answer:
Y alone can complete the job in **12 days**.