X can do half and Y can do one-third of a certain work in 5 and 10 days, respectively. Working together. in how many days will they be able to complete double of the work?

Options

A. `16(1)/2`
B. `7(1)/2`
C. `15`
D. 16

Answer

To solve the problem step by step, we will first determine the work done by X and Y individually, then calculate their combined efficiency, and finally find out how long it will take them to complete double the work together.

### Step 1: Determine the work done by X and Y

- **X can do half of the work in 5 days.**
Therefore, the total work (W) can be calculated as:
\[ \text{Total Work} = 2 \times \text{Work done by X} = 2 \times \frac{1}{2} = 1 \text{ (whole work)} \]
Thus, X can complete the whole work in:
\[ \text{Time taken by X} = 5 \text{ days} \times 2 = 10 \text{ days} \]

- **Y can do one-third of the work in 10 days.**
Therefore, the total work can be calculated as:
\[ \text{Total Work} = 3 \times \text{Work done by Y} = 3 \times \frac{1}{3} = 1 \text{ (whole work)} \]
Thus, Y can complete the whole work in:
\[ \text{Time taken by Y} = 10 \text{ days} \times 3 = 30 \text{ days} \]

### Step 2: Calculate the efficiency of X and Y

- **Efficiency of X:**
\[ \text{Efficiency of X} = \frac{1 \text{ (whole work)}}{10 \text{ days}} = \frac{1}{10} \text{ work/day} \]

- **Efficiency of Y:**
\[ \text{Efficiency of Y} = \frac{1 \text{ (whole work)}}{30 \text{ days}} = \frac{1}{30} \text{ work/day} \]

### Step 3: Calculate the combined efficiency of X and Y

- **Combined Efficiency:**
\[ \text{Combined Efficiency} = \text{Efficiency of X} + \text{Efficiency of Y} = \frac{1}{10} + \frac{1}{30} \]
To add these fractions, we need a common denominator:
\[ \text{LCM of 10 and 30} = 30 \]
Thus,
\[ \frac{1}{10} = \frac{3}{30} \quad \text{and} \quad \frac{1}{30} = \frac{1}{30} \]
Therefore,
\[ \text{Combined Efficiency} = \frac{3}{30} + \frac{1}{30} = \frac{4}{30} = \frac{2}{15} \text{ work/day} \]

### Step 4: Calculate the time taken to complete double the work

- **Double the work:**
Since the total work is 1, double the work is:
\[ \text{Double Work} = 2 \text{ (whole work)} \]

- **Time taken to complete double the work:**
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{2}{\frac{2}{15}} = 2 \times \frac{15}{2} = 15 \text{ days} \]

### Final Answer:
Thus, X and Y together will take **15 days** to complete double the work.

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