X and Y can do a piece of work in 30 days. They work together for 6 days and thenX quits and Y finishes the work in 32 days . In how many days can Y do the piece of work alone ?

Options

A. 30 days
B. 32 days
C. 34 days
D. 40 days

Answer

To solve the problem, we need to determine how many days Y can complete the work alone. Let's break down the solution step by step.

### Step 1: Determine the work done by X and Y together
Given that X and Y can complete the work together in 30 days, we can express their combined work rate as:
\[ \text{Work done in 1 day by X and Y} = \frac{1}{30} \text{ of the total work (W)} \]
Let the work done by X in one day be \( x \) and the work done by Y in one day be \( y \). Therefore, we have:
\[ x + y = \frac{W}{30} \tag{1} \]

### Step 2: Calculate the work done in 6 days
When X and Y work together for 6 days, the amount of work they complete is:
\[ \text{Work done in 6 days} = 6(x + y) = 6 \cdot \frac{W}{30} = \frac{W}{5} \tag{2} \]

### Step 3: Determine the remaining work after 6 days
After working together for 6 days, the remaining work is:
\[ \text{Remaining work} = W - \frac{W}{5} = \frac{4W}{5} \tag{3} \]

### Step 4: Calculate the work done by Y alone
After 6 days, X quits, and Y completes the remaining work in 32 days. The work done by Y in 32 days is:
\[ \text{Work done by Y in 32 days} = 32y \]
Since this work equals the remaining work, we have:
\[ 32y = \frac{4W}{5} \tag{4} \]

### Step 5: Relate Y's work rate to the total work
From equation (4), we can express \( y \) in terms of \( W \):
\[ y = \frac{4W}{5 \cdot 32} = \frac{W}{40} \tag{5} \]

### Step 6: Substitute \( y \) back into equation (1)
Now we substitute \( y \) from equation (5) into equation (1):
\[ x + \frac{W}{40} = \frac{W}{30} \]
To solve for \( x \), we need a common denominator. The least common multiple of 30 and 40 is 120. Thus, we rewrite the equation:
\[ x + \frac{3W}{120} = \frac{4W}{120} \]
Subtracting \( \frac{3W}{120} \) from both sides gives:
\[ x = \frac{4W}{120} - \frac{3W}{120} = \frac{W}{120} \tag{6} \]

### Step 7: Find the total work in terms of Y's work rate
Now we know both \( x \) and \( y \):
- \( x = \frac{W}{120} \)
- \( y = \frac{W}{40} \)

### Step 8: Determine how many days Y takes to complete the work alone
Let \( d \) be the number of days Y takes to complete the work alone. Then:
\[ d \cdot y = W \]
Substituting \( y \) from equation (5):
\[ d \cdot \frac{W}{40} = W \]
Dividing both sides by \( W \) (assuming \( W \neq 0 \)):
\[ \frac{d}{40} = 1 \implies d = 40 \]

### Final Answer
Y can complete the piece of work alone in **40 days**.

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