A work can be completed by A,B and C when working together in 12 days. If B working alone can completes 25% of the same work in 8 days and ratio of efficiency of C to that of A is 1:4. Find in how many days A and B can complete the same work together? (in days)

Options

A. `12(5)/(7)`
B. `13(5)/(7)`
C. `14(5)/(7)`
D. `11(5)/(7)`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the total work done by A, B, and C together
Given that A, B, and C can complete the work together in 12 days, we can calculate the total work in terms of efficiency.

Total work = Efficiency × Time
Let the total work be W.
Since they complete the work in 12 days, we have:
\[ W = \text{Efficiency of (A + B + C)} \times 12 \]

### Step 2: Calculate B's efficiency
It is given that B can complete 25% of the work in 8 days.
25% of the work means \( \frac{1}{4} \) of W.
If B completes \( \frac{1}{4} \) of the work in 8 days, then the total work W can be calculated as follows:

B's efficiency = \( \frac{1/4 \text{ of W}}{8 \text{ days}} = \frac{W}{32} \)
Thus, B's efficiency = \( \frac{W}{32} \)

### Step 3: Calculate the efficiency of A + B + C
Since A, B, and C together complete the work in 12 days, their combined efficiency is:
\[ \text{Efficiency of (A + B + C)} = \frac{W}{12} \]

### Step 4: Calculate the efficiency of A + C
Using the efficiency of B calculated in Step 2, we can find the efficiency of A + C:
\[ \text{Efficiency of (A + C)} = \text{Efficiency of (A + B + C)} - \text{Efficiency of B} \]
\[ \text{Efficiency of (A + C)} = \frac{W}{12} - \frac{W}{32} \]

To perform this subtraction, we need a common denominator. The LCM of 12 and 32 is 96.
Converting both fractions:
\[ \frac{W}{12} = \frac{8W}{96} \]
\[ \frac{W}{32} = \frac{3W}{96} \]
Now, subtract:
\[ \text{Efficiency of (A + C)} = \frac{8W}{96} - \frac{3W}{96} = \frac{5W}{96} \]

### Step 5: Determine the ratio of efficiencies of A and C
It is given that the ratio of the efficiency of C to A is 1:4.
Let the efficiency of C be x. Then the efficiency of A will be 4x.
Thus, we can write:
\[ \text{Efficiency of (A + C)} = 4x + x = 5x \]

From Step 4, we have:
\[ 5x = \frac{5W}{96} \]
Thus, we can find x:
\[ x = \frac{W}{96} \]
So, the efficiencies are:
- Efficiency of A = \( 4x = \frac{4W}{96} = \frac{W}{24} \)
- Efficiency of C = \( x = \frac{W}{96} \)

### Step 6: Calculate the efficiency of A + B
Now, we can find the combined efficiency of A and B:
\[ \text{Efficiency of (A + B)} = \text{Efficiency of A} + \text{Efficiency of B} \]
\[ \text{Efficiency of (A + B)} = \frac{W}{24} + \frac{W}{32} \]

Again, we need a common denominator (LCM of 24 and 32 is 96):
\[ \frac{W}{24} = \frac{4W}{96} \]
\[ \frac{W}{32} = \frac{3W}{96} \]
Now, adding:
\[ \text{Efficiency of (A + B)} = \frac{4W}{96} + \frac{3W}{96} = \frac{7W}{96} \]

### Step 7: Calculate the time taken by A and B to complete the work together
Now, we can find the time taken by A and B to complete the work:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of (A + B)}} \]
\[ \text{Time} = \frac{W}{\frac{7W}{96}} = \frac{96}{7} \]

Thus, the time taken by A and B to complete the work together is:
\[ \text{Time} = 13 \frac{5}{7} \text{ days} \]

### Final Answer
A and B can complete the work together in **13 days and 5/7 days**.

---