A can do `(1)/(4)` of a work in 5 days and B can do `(1)/(3)` of the same work in 10 days. Find the number of days in which both working together will complete the work.

Answer

To solve the problem step by step, we will determine how long A and B will take to complete the work when they work together.

### Step 1: Determine the total work done by A
A can do \( \frac{1}{4} \) of the work in 5 days. To find out how long A will take to complete the entire work, we can set up the equation:
\[ \text{Total work} = \text{Work done in 5 days} \times \text{Number of days to complete the work} \]
Since A does \( \frac{1}{4} \) of the work in 5 days, the total work can be calculated as:
\[ \text{Total work} = 5 \text{ days} \times 4 = 20 \text{ days} \]

### Step 2: Determine A's work rate
Now, we can find A's work rate (work done in one day):
\[ \text{A's work rate} = \frac{1 \text{ work}}{20 \text{ days}} = \frac{1}{20} \text{ work per day} \]

### Step 3: Determine the total work done by B
B can do \( \frac{1}{3} \) of the work in 10 days. Similarly, we calculate how long B will take to complete the entire work:
\[ \text{Total work} = 10 \text{ days} \times 3 = 30 \text{ days} \]

### Step 4: Determine B's work rate
Now, we can find B's work rate:
\[ \text{B's work rate} = \frac{1 \text{ work}}{30 \text{ days}} = \frac{1}{30} \text{ work per day} \]

### Step 5: Combine A's and B's work rates
To find the combined work rate of A and B when they work together, we add their individual work rates:
\[ \text{Combined work rate} = \text{A's work rate} + \text{B's work rate} = \frac{1}{20} + \frac{1}{30} \]

### Step 6: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple of 20 and 30 is 60. We convert each fraction:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
Now we can add them:
\[ \text{Combined work rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]

### Step 7: Calculate the time taken to complete the work together
The combined work rate of \( \frac{1}{12} \) means that A and B together can complete \( \frac{1}{12} \) of the work in one day. Therefore, the total time taken to complete the work together is:
\[ \text{Time taken} = \frac{1 \text{ work}}{\frac{1}{12} \text{ work per day}} = 12 \text{ days} \]

### Final Answer
A and B together will complete the work in **12 days**.

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