A can do a piece of work in 14 days. Other person B is `20%` more efficient than A. In how many days will A and B together complete the same piece of work?

Options

A. `9 10/11 days`
B. `10 10/11 days`
C. `6 4/11 days`
D. `12 10/11 day`

Answer

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

### Step 1: Determine A's Efficiency
A can complete the work in 14 days. Therefore, A's efficiency can be calculated as:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{14 \text{ days}} = \frac{1}{14} \]

### Step 2: Determine B's Efficiency
B is 20% more efficient than A. To find B's efficiency, we first calculate 20% of A's efficiency:
\[ 20\% \text{ of A's efficiency} = 0.2 \times \frac{1}{14} = \frac{0.2}{14} = \frac{1}{70} \]
Now, we add this to A's efficiency to find B's efficiency:
\[ \text{Efficiency of B} = \text{Efficiency of A} + 20\% \text{ of A's efficiency} = \frac{1}{14} + \frac{1}{70} \]
To add these fractions, we need a common denominator. The least common multiple of 14 and 70 is 70:
\[ \frac{1}{14} = \frac{5}{70} \]
Now we can add:
\[ \text{Efficiency of B} = \frac{5}{70} + \frac{1}{70} = \frac{6}{70} = \frac{3}{35} \]

### Step 3: Calculate Combined Efficiency of A and B
Now that we have the efficiencies of both A and B, we can find their combined efficiency:
\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{14} + \frac{3}{35} \]
Again, we need a common denominator. The least common multiple of 14 and 35 is 70:
\[ \frac{1}{14} = \frac{5}{70}, \quad \frac{3}{35} = \frac{6}{70} \]
Now we can add:
\[ \text{Combined Efficiency} = \frac{5}{70} + \frac{6}{70} = \frac{11}{70} \]

### Step 4: Calculate Time Taken by A and B Together
The total work can be represented as 1 unit of work. To find the time taken by A and B together to complete the work, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{1}{\frac{11}{70}} = \frac{70}{11} \]
This means A and B together will take \(\frac{70}{11}\) days to complete the work.

### Final Answer
Thus, A and B together will complete the work in \(6 \frac{4}{11}\) days.

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