P, Q and R together can complete a work in 40 days. P and Q together can complete the same work in 90 days. Then R alone can complete the same work in how many days?

Options

A. 1800
B. 900
C. 1200
D. 1500

Answer

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

1. **Understanding the Work Rates**:
- Let the total work be represented as 1 unit of work.
- If P, Q, and R together can complete the work in 40 days, their combined work rate is:
\[ \text{Work Rate of P + Q + R} = \frac{1}{40} \text{ (units of work per day)} \]

2. **Calculating the Work Rate of P and Q**:
- If P and Q together can complete the work in 90 days, their combined work rate is:
\[ \text{Work Rate of P + Q} = \frac{1}{90} \text{ (units of work per day)} \]

3. **Finding the Work Rate of R**:
- To find R's work rate, we can subtract the work rate of P and Q from the work rate of P, Q, and R:
\[ \text{Work Rate of R} = \text{Work Rate of P + Q + R} - \text{Work Rate of P + Q} \]
\[ \text{Work Rate of R} = \frac{1}{40} - \frac{1}{90} \]

4. **Finding a Common Denominator**:
- The least common multiple (LCM) of 40 and 90 is 360. We will convert both fractions to have a common denominator:
\[ \frac{1}{40} = \frac{9}{360} \quad \text{(since } 1 \times 9 = 9 \text{ and } 40 \times 9 = 360\text{)} \]
\[ \frac{1}{90} = \frac{4}{360} \quad \text{(since } 1 \times 4 = 4 \text{ and } 90 \times 4 = 360\text{)} \]

5. **Performing the Subtraction**:
- Now we can subtract the two fractions:
\[ \text{Work Rate of R} = \frac{9}{360} - \frac{4}{360} = \frac{5}{360} \]

6. **Calculating the Time Taken by R Alone**:
- To find out how many days R alone will take to complete the work, we can use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Work Rate of R}} = \frac{1}{\frac{5}{360}} = \frac{360}{5} = 72 \text{ days} \]

Thus, R alone can complete the work in **72 days**.