The efficiency of A is 40% more than that of B and efficiency of C is `14(2)/(7)%` less than that of A. Time taken by A and B together to complete the work is 11 days. Find the number of days taken by all of them to complete the thrice of the same work, working simultaneously.

Options

A. A)13 days
B. B)22 days
C. C)11 days
D. D)26 days

Answer

To solve the problem step by step, we will first determine the efficiencies of A, B, and C based on the information provided, then calculate the total work done, and finally find out how long it will take for A, B, and C to complete three times that work together.

### Step 1: Determine the efficiencies of A, B, and C

Let the efficiency of B be \( x \).
Then, the efficiency of A, which is 40% more than B, can be calculated as:
\[ \text{Efficiency of A} = x + 0.4x = 1.4x \]

Next, we need to find the efficiency of C. The efficiency of C is given as \( 14 \frac{2}{7}\% \) less than that of A.
First, convert \( 14 \frac{2}{7}\% \) to a fraction:
\[ 14 \frac{2}{7}\% = \frac{100}{7}\% \]
This means C's efficiency is:
\[ \text{Efficiency of C} = \text{Efficiency of A} - \left(\frac{100}{7} \times \frac{\text{Efficiency of A}}{100}\right) \]
\[ = 1.4x - \frac{1.4x}{7} = 1.4x \left(1 - \frac{1}{7}\right) = 1.4x \times \frac{6}{7} = \frac{8.4x}{7} \]

Thus, we have:
- Efficiency of A = \( 1.4x \)
- Efficiency of B = \( x \)
- Efficiency of C = \( \frac{8.4x}{7} \)

### Step 2: Calculate the combined efficiency of A and B

The combined efficiency of A and B is:
\[ \text{Efficiency of A + B} = 1.4x + x = 2.4x \]

### Step 3: Calculate the total work done by A and B in 11 days

According to the problem, A and B together take 11 days to complete the work. Therefore, the total work can be calculated as:
\[ \text{Total Work} = \text{Efficiency of A + B} \times \text{Time} = 2.4x \times 11 = 26.4x \]

### Step 4: Calculate the total work for three times the work

Now, we need to find the time taken by A, B, and C to complete three times the work:
\[ \text{Total Work for 3 times} = 3 \times 26.4x = 79.2x \]

### Step 5: Calculate the combined efficiency of A, B, and C

The combined efficiency of A, B, and C is:
\[ \text{Efficiency of A + B + C} = 1.4x + x + \frac{8.4x}{7} \]
To combine these, we can convert \( 1.4x \) and \( x \) into fractions with a common denominator of 7:
\[ 1.4x = \frac{9.8x}{7}, \quad x = \frac{7x}{7} \]
Thus,
\[ \text{Efficiency of A + B + C} = \frac{9.8x}{7} + \frac{7x}{7} + \frac{8.4x}{7} = \frac{25.2x}{7} \]

### Step 6: Calculate the time taken by A, B, and C to complete the work

Now, we can find the time taken by A, B, and C to complete the total work of \( 79.2x \):
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of A + B + C}} = \frac{79.2x}{\frac{25.2x}{7}} = 79.2x \times \frac{7}{25.2x} \]
The \( x \) cancels out:
\[ = \frac{79.2 \times 7}{25.2} = \frac{554.4}{25.2} = 22 \text{ days} \]

### Final Answer
The time taken by A, B, and C to complete thrice the work together is **22 days**.

---