P and Q together can complete a work in 20 days. If P alone can complete the same work in 36 days, then in how many days Q alone can complete the same work?

Options

A. 48 days
B. 42 days
C. 45 days
D. 51 days

Answer

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

### Step 1: Determine the work done by P and Q together
P and Q together can complete the work in 20 days. Therefore, their combined work rate per day is:
\[ \text{Work rate of P and Q together} = \frac{1}{20} \text{ (work per day)} \]

### Step 2: Determine the work done by P alone
P can complete the work alone in 36 days. Thus, P's work rate per day is:
\[ \text{Work rate of P} = \frac{1}{36} \text{ (work per day)} \]

### Step 3: Set up the equation for Q's work rate
Let Q's work rate be represented as \( \frac{1}{X} \), where \( X \) is the number of days Q takes to complete the work alone. According to the problem, the combined work rates of P and Q can be expressed as:
\[ \text{Work rate of P} + \text{Work rate of Q} = \text{Work rate of P and Q together} \]
This can be written as:
\[ \frac{1}{36} + \frac{1}{X} = \frac{1}{20} \]

### Step 4: Solve for Q's work rate
To solve for \( \frac{1}{X} \), we rearrange the equation:
\[ \frac{1}{X} = \frac{1}{20} - \frac{1}{36} \]

### Step 5: Find a common denominator
The least common multiple (LCM) of 20 and 36 is 180. We convert the fractions:
\[ \frac{1}{20} = \frac{9}{180} \quad \text{and} \quad \frac{1}{36} = \frac{5}{180} \]
Thus, we have:
\[ \frac{1}{X} = \frac{9}{180} - \frac{5}{180} = \frac{4}{180} \]

### Step 6: Simplify the fraction
Now simplify \( \frac{4}{180} \):
\[ \frac{4}{180} = \frac{1}{45} \]

### Step 7: Solve for X
From \( \frac{1}{X} = \frac{1}{45} \), we find:
\[ X = 45 \]

### Conclusion
Q alone can complete the work in **45 days**.

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