A can do a work in 15 days. B is `25%` more effcient than a In how many days working together A and B is complete the same work ?

Options

A. `(21)/(4)`
B. `(25)/(7)`
C. `(24)/(5)`
D. `(20)/(3)`

Answer

To solve the problem step-by-step, we will follow these calculations:

### Step 1: Determine A's Efficiency
A can complete the work in 15 days. The efficiency of A can be calculated as:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{15 \text{ days}} = \frac{1}{15} \]

### Step 2: Calculate B's Efficiency
B is 25% more efficient than A. To find B's efficiency, we first convert 25% into a fraction:
\[ 25\% = \frac{25}{100} = \frac{1}{4} \]
This means B's efficiency is:
\[ \text{Efficiency of B} = \text{Efficiency of A} + 25\% \text{ of Efficiency of A} = \frac{1}{15} + \frac{1}{4} \times \frac{1}{15} \]
To combine these, we need a common denominator. The least common multiple of 15 and 4 is 60:
\[ \text{Efficiency of A} = \frac{4}{60}, \quad \text{Efficiency of B} = \frac{15}{60} \]
So,
\[ \text{Efficiency of B} = \frac{4}{60} + \frac{1}{4} \times \frac{4}{60} = \frac{4}{60} + \frac{1}{15} = \frac{4}{60} + \frac{4}{60} = \frac{8}{60} = \frac{2}{15} \]

### Step 3: Calculate Total Efficiency
Now, we can find the total efficiency of A and B working together:
\[ \text{Total Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{4}{60} + \frac{8}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 4: Calculate Total Work
The total work can be calculated as:
\[ \text{Total Work} = \text{Efficiency of A} \times \text{Days taken by A} = \frac{1}{15} \times 15 = 1 \text{ work unit} \]

### Step 5: Calculate Time Taken by A and B Together
Now, we can find the time taken by A and B together to complete the work:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Final Answer
Thus, A and B together will complete the work in **5 days**.

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