Sunny and Paula can complete a piece of work together in 12 hours. While Paula takes 18 hours to complete the work 2 alone, another individual, Jane, takes 45 hours to complete the same piece of work alone. How many hours would it take Sunny and Jane to complete the same work together ?

Options

A. 18
B. 16
C. 15
D. 20

Answer

To solve the problem step by step, we will first determine the work rates of Sunny, Paula, and Jane, and then find out how long it would take Sunny and Jane to complete the work together.

### Step 1: Determine Paula's work rate
Paula can complete the work alone in 18 hours. Therefore, her work rate is:
\[ \text{Paula's work rate} = \frac{1}{18} \text{ (work per hour)} \]

### Step 2: Determine Sunny and Paula's combined work rate
Sunny and Paula can complete the work together in 12 hours. Therefore, their combined work rate is:
\[ \text{Sunny and Paula's combined work rate} = \frac{1}{12} \text{ (work per hour)} \]

### Step 3: Set up the equation for Sunny's work rate
Let Sunny's work rate be \( \frac{1}{x} \). According to the information provided:
\[ \frac{1}{x} + \frac{1}{18} = \frac{1}{12} \]

### Step 4: Solve for Sunny's work rate
To solve for \( \frac{1}{x} \), we can rearrange the equation:
\[ \frac{1}{x} = \frac{1}{12} - \frac{1}{18} \]

To subtract these fractions, we need a common denominator. The least common multiple of 12 and 18 is 36. Thus, we convert the fractions:
\[ \frac{1}{12} = \frac{3}{36}, \quad \frac{1}{18} = \frac{2}{36} \]
Now substituting back into the equation:
\[ \frac{1}{x} = \frac{3}{36} - \frac{2}{36} = \frac{1}{36} \]

### Step 5: Calculate Sunny's time to complete the work alone
From \( \frac{1}{x} = \frac{1}{36} \), we find:
\[ x = 36 \]
So, Sunny can complete the work alone in 36 hours.

### Step 6: Determine Jane's work rate
Jane can complete the work alone in 45 hours. Therefore, her work rate is:
\[ \text{Jane's work rate} = \frac{1}{45} \text{ (work per hour)} \]

### Step 7: Calculate Sunny and Jane's combined work rate
Now we can find the combined work rate of Sunny and Jane:
\[ \text{Sunny and Jane's combined work rate} = \frac{1}{36} + \frac{1}{45} \]

### Step 8: Find a common denominator
The least common multiple of 36 and 45 is 180. We convert the fractions:
\[ \frac{1}{36} = \frac{5}{180}, \quad \frac{1}{45} = \frac{4}{180} \]
Now adding these fractions:
\[ \text{Combined work rate} = \frac{5}{180} + \frac{4}{180} = \frac{9}{180} = \frac{1}{20} \]

### Step 9: Calculate the time taken by Sunny and Jane together
If their combined work rate is \( \frac{1}{20} \), then the time taken to complete the work together is:
\[ \text{Time} = 20 \text{ hours} \]

### Final Answer
Sunny and Jane would take **20 hours** to complete the work together.

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