Ratio of efficiency of A and B in completing a work is 3 : 4. Both started to work together but A left after 2 days. Another person C joins B and they together complete the remaining work in 6 days. If A and B together can complete the work in 8 days then C alone can complete the work in how many days?

Options

A. A)`(27)/(4)`days
B. B)`(56)/(3)`days
C. C)`(41)/(3)`days
D. D)`(28)/(3)`days

Answer

To solve the problem step by step, let's break down the information given and find the required solution.

### Step 1: Understand the Efficiency Ratio
The efficiency ratio of A and B is given as 3:4. This means:
- Let the efficiency of A = 3 units
- Let the efficiency of B = 4 units

### Step 2: Calculate Total Efficiency of A and B
The combined efficiency of A and B when they work together is:
\[ \text{Efficiency of A + Efficiency of B} = 3 + 4 = 7 \text{ units} \]

### Step 3: Calculate Total Work
It is given that A and B together can complete the work in 8 days. Therefore, the total work can be calculated as:
\[ \text{Total Work} = \text{Combined Efficiency} \times \text{Time} = 7 \text{ units} \times 8 \text{ days} = 56 \text{ units} \]

### Step 4: Work Done by A and B in 2 Days
When A and B work together for 2 days, the amount of work done is:
\[ \text{Work Done} = \text{Combined Efficiency} \times \text{Time} = 7 \text{ units} \times 2 \text{ days} = 14 \text{ units} \]

### Step 5: Remaining Work
The remaining work after A leaves is:
\[ \text{Remaining Work} = \text{Total Work} - \text{Work Done} = 56 \text{ units} - 14 \text{ units} = 42 \text{ units} \]

### Step 6: Work Done by B and C Together
B continues to work and is joined by C. They complete the remaining work in 6 days. The combined efficiency of B and C is:
Let the efficiency of C be \( c \) units. Then:
\[ \text{Efficiency of B + Efficiency of C} = 4 + c \]

The work done by B and C together in 6 days is:
\[ \text{Work Done} = \text{Combined Efficiency} \times \text{Time} = (4 + c) \times 6 \]

Setting this equal to the remaining work:
\[ (4 + c) \times 6 = 42 \]

### Step 7: Solve for C's Efficiency
Expanding the equation:
\[ 24 + 6c = 42 \]

Now, isolate \( c \):
\[ 6c = 42 - 24 \]
\[ 6c = 18 \]
\[ c = 3 \text{ units} \]

### Step 8: Calculate Time Taken by C Alone
Now we need to find out how many days C will take to complete the entire work alone. The time taken by C to complete the work is given by:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of C}} = \frac{56 \text{ units}}{3 \text{ units}} = \frac{56}{3} \text{ days} \]

### Final Answer
C alone can complete the work in \( \frac{56}{3} \) days.

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