The ratio of efficiency of A is to C is.5 :3 The ratio of number of days taken by B is to C is 2:3 . A takes 6 days less than C, when A and C completes the work individually. B and C started the work arid left after 3 days. The number of days taken by A to finish the remaining work is:

Options

A. 4.5
B. 5
C. 6
D. `9 1/3`

Answer

To solve the problem step by step, we will first analyze the given information and then calculate the required values.

### Step 1: Understand the Ratios of Efficiencies
The ratio of efficiencies of A to C is given as 5:3. This means:
- Efficiency of A = 5x
- Efficiency of C = 3x

### Step 2: Understand the Days Taken by B and C
The ratio of the number of days taken by B to C is given as 2:3. Let the number of days taken by B be 2y and by C be 3y.

### Step 3: Relate Days Taken by A and C
It is given that A takes 6 days less than C. Therefore, we can write:
- Days taken by A = Days taken by C - 6
- Days taken by A = 3y - 6

### Step 4: Set Up the Equation for A's Days
Since we know A's efficiency is 5x and the time taken by A is (3y - 6), we can express the total work done as:
- Total Work = Efficiency × Time
- Total Work = 5x × (3y - 6)

### Step 5: Calculate Total Work Using C's Days
For C, the total work can also be expressed as:
- Total Work = Efficiency × Time
- Total Work = 3x × 3y = 9xy

### Step 6: Equate the Total Work Expressions
From the above expressions, we have:
\[ 5x(3y - 6) = 9xy \]

### Step 7: Simplify the Equation
Expanding the left side:
\[ 15xy - 30x = 9xy \]
Now, rearranging gives:
\[ 15xy - 9xy = 30x \]
\[ 6xy = 30x \]

### Step 8: Solve for y
Dividing both sides by x (assuming x ≠ 0):
\[ 6y = 30 \]
\[ y = 5 \]

### Step 9: Calculate Days Taken by A, B, and C
Now we can find the number of days taken by each:
- Days taken by C = 3y = 3(5) = 15 days
- Days taken by B = 2y = 2(5) = 10 days
- Days taken by A = 3y - 6 = 15 - 6 = 9 days

### Step 10: Calculate Total Work
Total work can be calculated using C's days:
- Total Work = 3x × 15 = 45x

### Step 11: Calculate Work Done by B and C in 3 Days
Now, B and C worked together for 3 days:
- Work done by B in 3 days = Efficiency of B × Days = (3/2) × 3 = 4.5 units
- Work done by C in 3 days = Efficiency of C × Days = (3) × 3 = 9 units
- Total work done by B and C in 3 days = 4.5 + 9 = 13.5 units

### Step 12: Calculate Remaining Work
Remaining work after 3 days:
- Remaining Work = Total Work - Work Done = 45 - 13.5 = 31.5 units

### Step 13: Calculate Days Taken by A to Finish Remaining Work
Now, A will complete the remaining work:
- A's efficiency = 5x = 5(3) = 15 units/day
- Days taken by A to finish remaining work = Remaining Work / A's Efficiency = 31.5 / 15 = 2.1 days

### Final Answer
The number of days taken by A to finish the remaining work is **2.1 days**.