A is 40 % more efficient than B. If B can complete this work in 24 days. In how many days will both A and B complete this work ?

Answer

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

### Step 1: Determine the efficiency of B
Given that B can complete the work in 24 days, we can find B's efficiency. The efficiency of B can be calculated as:

\[ \text{Efficiency of B} = \frac{1}{\text{Time taken by B}} = \frac{1}{24} \text{ work/day} \]

### Step 2: Determine the efficiency of A
Since A is 40% more efficient than B, we can express A's efficiency in terms of B's efficiency.

\[ \text{Efficiency of A} = \text{Efficiency of B} + 40\% \text{ of Efficiency of B} \]

Calculating 40% of B's efficiency:

\[ 40\% \text{ of } \frac{1}{24} = \frac{40}{100} \times \frac{1}{24} = \frac{2}{24} = \frac{1}{12} \]

Now, we can find A's efficiency:

\[ \text{Efficiency of A} = \frac{1}{24} + \frac{1}{12} \]

To add these fractions, we need a common denominator. The least common multiple of 24 and 12 is 24.

\[ \text{Efficiency of A} = \frac{1}{24} + \frac{2}{24} = \frac{3}{24} = \frac{1}{8} \text{ work/day} \]

### Step 3: Calculate the combined efficiency of A and B
Now, we can find the combined efficiency of A and B:

\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{8} + \frac{1}{24} \]

Finding a common denominator (which is 24):

\[ \text{Combined Efficiency} = \frac{3}{24} + \frac{1}{24} = \frac{4}{24} = \frac{1}{6} \text{ work/day} \]

### Step 4: Calculate the time taken by A and B together to complete the work
To find the time taken by A and B together to complete the work, we take the reciprocal of their combined efficiency:

\[ \text{Time taken} = \frac{1}{\text{Combined Efficiency}} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \]

### Final Answer
Both A and B together will complete the work in **6 days**.

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