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. 40
B. 72
C. 84
D. 90

Answer

To solve the problem, we need to find out how many days R alone can complete the work. We'll break down the solution step by step.

### Step 1: Determine the work done by P, Q, and R together in one day.
Given that P, Q, and R together can complete the work in 40 days, the work done by them in one day is:
\[ \text{Work done by P, Q, and R in one day} = \frac{1}{40} \]

### Step 2: Determine the work done by P and Q together in one day.
It is given that P and Q together can complete the work in 90 days. Therefore, the work done by P and Q in one day is:
\[ \text{Work done by P and Q in one day} = \frac{1}{90} \]

### Step 3: Calculate the work done by R in one day.
To find the work done by R alone in one day, we subtract the work done by P and Q from the work done by P, Q, and R:
\[ \text{Work done by R in one day} = \text{Work done by P, Q, and R in one day} - \text{Work done by P and Q in one day} \]
\[ \text{Work done by R in one day} = \frac{1}{40} - \frac{1}{90} \]

### Step 4: Find a common denominator and perform the subtraction.
The least common multiple (LCM) of 40 and 90 is 360. We convert both fractions:
\[ \frac{1}{40} = \frac{9}{360} \quad \text{(since } 1 \times 9 = 9 \text{ and } 40 \times 9 = 360\text{)} \]
\[ \frac{1}{90} = \frac{4}{360} \quad \text{(since } 1 \times 4 = 4 \text{ and } 90 \times 4 = 360\text{)} \]
Now we can subtract:
\[ \text{Work done by R in one day} = \frac{9}{360} - \frac{4}{360} = \frac{5}{360} \]

### Step 5: Calculate the number of days R takes to complete the work alone.
If R does \(\frac{5}{360}\) of the work in one day, then the total time R takes to complete the work alone is:
\[ \text{Time taken by R} = \frac{1}{\text{Work done by R in one day}} = \frac{1}{\frac{5}{360}} = \frac{360}{5} = 72 \text{ days} \]

### Final Answer:
R alone can complete the work in **72 days**.

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