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 both A and B together can do the work ?

Options

A. `7 3/4`
B. `8 4/5`
C. `9 3/8`
D. 10

Answer

To solve the problem step by step, we will first determine how long A and B take to complete the entire work individually, and then we will find out how long they take together.

### Step 1: Calculate the time taken by A to complete the entire work.
A can do \( \frac{1}{3} \) of the work in 5 days. To find out how many days A will take to complete the entire work, we can set up the following equation:

\[ \text{Time taken by A} = \text{Total work} \div \text{Work done by A in 1 day} \]

First, we find out how much work A does in one day:

\[ \text{Work done by A in 1 day} = \frac{1}{3} \div 5 = \frac{1}{15} \]

Now, we can calculate the total time taken by A to complete the whole work:

\[ \text{Time taken by A} = 1 \div \frac{1}{15} = 15 \text{ days} \]

### Step 2: Calculate the time taken by B to complete the entire work.
B can do \( \frac{2}{5} \) of the work in 10 days. Similarly, we find out how much work B does in one day:

\[ \text{Work done by B in 1 day} = \frac{2}{5} \div 10 = \frac{1}{25} \]

Now, we can calculate the total time taken by B to complete the whole work:

\[ \text{Time taken by B} = 1 \div \frac{1}{25} = 25 \text{ days} \]

### Step 3: Calculate the combined work done by A and B in one day.
To find out how much work A and B can do together in one day, we add their individual work rates:

\[ \text{Combined work in 1 day} = \text{Work done by A in 1 day} + \text{Work done by B in 1 day} \]

Substituting the values we found:

\[ \text{Combined work in 1 day} = \frac{1}{15} + \frac{1}{25} \]

### Step 4: Find the least common multiple (LCM) to add the fractions.
The LCM of 15 and 25 is 75. Now we convert the fractions:

\[ \frac{1}{15} = \frac{5}{75}, \quad \frac{1}{25} = \frac{3}{75} \]

Now we can add the fractions:

\[ \text{Combined work in 1 day} = \frac{5}{75} + \frac{3}{75} = \frac{8}{75} \]

### Step 5: Calculate the total time taken by A and B together to complete the work.
To find out how many days A and B together will take to complete the work, we take the reciprocal of their combined work rate:

\[ \text{Time taken by A and B together} = 1 \div \frac{8}{75} = \frac{75}{8} \text{ days} \]

### Step 6: Convert the fraction to a mixed number.
Now, we can convert \( \frac{75}{8} \) into a mixed number:

\[ \frac{75}{8} = 9 \frac{3}{8} \text{ days} \]

### Final Answer:
Thus, A and B together can complete the work in \( 9 \frac{3}{8} \) days.

---