A can do 1/3 of a work in 5 days and B can do 2/5 of the work in 10 days. In how many days can both A and B together do the work?

Options

A. `7^3//4` days
B. `8^(4//5)` days
C. `9^(3//8)` days
D. 10 days

Answer

To solve the problem of how many days A and B together can complete the work, we will follow these steps:

### Step 1: Determine the work done by A in one day.
A can do \( \frac{1}{3} \) of the work in 5 days. Therefore, in one day, A can do:
\[ \text{Work done by A in one day} = \frac{1/3}{5} = \frac{1}{15} \]

### Step 2: Determine the work done by B in one day.
B can do \( \frac{2}{5} \) of the work in 10 days. Therefore, in one day, B can do:
\[ \text{Work done by B in one day} = \frac{2/5}{10} = \frac{2}{50} = \frac{1}{25} \]

### Step 3: Calculate the combined work done by A and B in one day.
Now, we add the work done by A and B in one day:
\[ \text{Combined work done in one day} = \frac{1}{15} + \frac{1}{25} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 25 is 75.

Now, convert each fraction:
\[ \frac{1}{15} = \frac{5}{75} \quad \text{and} \quad \frac{1}{25} = \frac{3}{75} \]

Now, add the fractions:
\[ \text{Combined work done in one day} = \frac{5}{75} + \frac{3}{75} = \frac{8}{75} \]

### Step 4: Calculate the total time taken by A and B together to complete the work.
If A and B together can do \( \frac{8}{75} \) of the work in one day, then the total time taken to complete the entire work (1 unit of work) is:
\[ \text{Total time} = \frac{1}{\text{Combined work done in one day}} = \frac{1}{\frac{8}{75}} = \frac{75}{8} \text{ days} \]

### Final Answer:
A and B together can complete the work in \( \frac{75}{8} \) days, which is equivalent to 9.375 days.

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