P can do a piece of work in 12 days , while Q alone can finish it in 8 days . With the help of R , they can finish the work in 4 days. How long willRtake to finish the work alone?

Options

A. 25 days
B. 34 days
C. 14 days
D. 24 days

Answer

To solve the problem step by step, let's first understand the work rates of P, Q, and R.

### Step 1: Calculate the work rates of P and Q.
- P can complete the work in 12 days. Therefore, P's work rate is:
\[ \text{Work rate of P} = \frac{1}{12} \text{ (work per day)} \]

- Q can complete the work in 8 days. Therefore, Q's work rate is:
\[ \text{Work rate of Q} = \frac{1}{8} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of P and Q.
- The combined work rate of P and Q is:
\[ \text{Combined work rate of P and Q} = \frac{1}{12} + \frac{1}{8} \]

To add these fractions, we need a common denominator. The least common multiple of 12 and 8 is 24.

- Converting the fractions:
\[ \frac{1}{12} = \frac{2}{24}, \quad \frac{1}{8} = \frac{3}{24} \]

- Now, adding them:
\[ \text{Combined work rate of P and Q} = \frac{2}{24} + \frac{3}{24} = \frac{5}{24} \]

### Step 3: Calculate the combined work rate of P, Q, and R.
- It is given that P, Q, and R together can finish the work in 4 days. Therefore, their combined work rate is:
\[ \text{Combined work rate of P, Q, and R} = \frac{1}{4} \]

### Step 4: Set up the equation to find R's work rate.
- We know:
\[ \text{Combined work rate of P, Q, and R} = \text{Combined work rate of P and Q} + \text{Work rate of R} \]
\[ \frac{1}{4} = \frac{5}{24} + \text{Work rate of R} \]

### Step 5: Solve for R's work rate.
- Rearranging the equation to find R's work rate:
\[ \text{Work rate of R} = \frac{1}{4} - \frac{5}{24} \]

To subtract these fractions, we convert \(\frac{1}{4}\) to have a denominator of 24:
\[ \frac{1}{4} = \frac{6}{24} \]

Now substituting:
\[ \text{Work rate of R} = \frac{6}{24} - \frac{5}{24} = \frac{1}{24} \]

### Step 6: Calculate the time R takes to finish the work alone.
- Since R's work rate is \(\frac{1}{24}\), it means R can complete the work in 24 days.

### Final Answer:
R will take **24 days** to finish the work alone.

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