P can do a piece of work in 15 days and Q in 20 days. With help of R, they finish the work in 5 days. In how many days R alone can do the same work?

Options

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

Answer

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

### Step 1: Determine the work done by P and Q in one day.

- P can complete the work in 15 days, so the work done by P in one day is:
\[ \text{Work done by P in one day} = \frac{1}{15} \text{ of the work} \]

- Q can complete the work in 20 days, so the work done by Q in one day is:
\[ \text{Work done by Q in one day} = \frac{1}{20} \text{ of the work} \]

### Step 2: Find the LCM of the days taken by P and Q.

- The LCM of 15 and 20 is 60. This means the total work can be considered as 60 units.

### Step 3: Calculate the work done by P and Q in units.

- Work done by P in one day:
\[ \text{Work done by P in one day} = \frac{60}{15} = 4 \text{ units} \]

- Work done by Q in one day:
\[ \text{Work done by Q in one day} = \frac{60}{20} = 3 \text{ units} \]

### Step 4: Calculate the total work done by P and Q in one day.

- The total work done by P and Q together in one day:
\[ \text{Total work done by P and Q in one day} = 4 + 3 = 7 \text{ units} \]

### Step 5: Determine the work done by P, Q, and R together.

- They finish the work in 5 days, which means:
\[ \text{Total work done by P, Q, and R in one day} = \frac{60}{5} = 12 \text{ units} \]

### Step 6: Calculate the work done by R in one day.

- Since P and Q together do 7 units of work in one day, the work done by R in one day is:
\[ \text{Work done by R in one day} = 12 - 7 = 5 \text{ units} \]

### Step 7: Calculate how many days R alone can do the work.

- If R does 5 units of work in one day, the number of days R will take to complete the total work (60 units) is:
\[ \text{Days taken by R} = \frac{60}{5} = 12 \text{ days} \]

### Final Answer:
R alone can complete the work in **12 days**.

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