A can do one-fourth of a work in 5 days and B can do two-fifth of the work in 10 days. In how many days can both A and B together do the work ?

Options

A. `12(1/9)`
B. `11(1/9)`
C. `13(1/9)`
D. `10(1/9)`

Answer

To solve the problem step by step, we need to determine how much work A and B can do individually and then combine their efforts to find out how long it will take for them to complete the work together.

### Step 1: Determine the total work done by A in one day.
A can do one-fourth of the work in 5 days.

- Work done by A in 5 days = \( \frac{1}{4} \)
- Therefore, work done by A in 1 day = \( \frac{1}{4} \div 5 = \frac{1}{20} \)

### Step 2: Determine the total work done by B in one day.
B can do two-fifths of the work in 10 days.

- Work done by B in 10 days = \( \frac{2}{5} \)
- Therefore, work done by B in 1 day = \( \frac{2}{5} \div 10 = \frac{2}{50} = \frac{1}{25} \)

### Step 3: Calculate the combined work done by A and B in one day.
Now, we can find the total work done by A and B together in one day.

- Work done by A in 1 day = \( \frac{1}{20} \)
- Work done by B in 1 day = \( \frac{1}{25} \)

To add these fractions, we need a common denominator. The least common multiple of 20 and 25 is 100.

- Convert \( \frac{1}{20} \) to a fraction with a denominator of 100:
\[ \frac{1}{20} = \frac{5}{100} \]
- Convert \( \frac{1}{25} \) to a fraction with a denominator of 100:
\[ \frac{1}{25} = \frac{4}{100} \]

Now, add the two fractions:
\[ \text{Total work done in 1 day} = \frac{5}{100} + \frac{4}{100} = \frac{9}{100} \]

### Step 4: Determine how many days A and B will take to complete the work together.
If A and B together can do \( \frac{9}{100} \) of the work in one day, then to complete the entire work (which is 1), we can set up the equation:

\[ \text{Days} = \frac{1}{\text{Work done in 1 day}} = \frac{1}{\frac{9}{100}} = \frac{100}{9} \]

### Final Answer:
A and B together can complete the work in \( \frac{100}{9} \) days, which is approximately 11.11 days.