Each of A, B and C need a certain unique time to do a certain work. C needs 1 hour less than A to complete the work. Working together, they require 30 minutes to complete 50% of the job. The work also gets completed if A and B start working together and A leaves after 1 hour and B works for a further 3 hours. How much work does C do per hour?

Options

A. `16.66%`
B. `33.33%`
C. `50%`
D. `66.66%`

Answer

To solve the problem, let's denote the time taken by A to complete the work as \( A \) hours. Since C takes 1 hour less than A, the time taken by C to complete the work will be \( A - 1 \) hours.

Let’s denote the time taken by B to complete the work as \( B \) hours.

### Step 1: Determine the work rates of A, B, and C
- The work rate of A is \( \frac{1}{A} \) (work done per hour).
- The work rate of B is \( \frac{1}{B} \).
- The work rate of C is \( \frac{1}{A-1} \).

### Step 2: Calculate the combined work rate of A and C
Since A and C together complete 50% of the work in 30 minutes (or 0.5 hours), we can set up the equation:
\[ \left( \frac{1}{A} + \frac{1}{A-1} \right) \times 0.5 = 0.5 \]

### Step 3: Simplify the equation
Multiplying both sides by 2 to eliminate the fraction:
\[ \frac{1}{A} + \frac{1}{A-1} = 1 \]

### Step 4: Find a common denominator and solve
The common denominator for the left-hand side is \( A(A-1) \):
\[ \frac{(A-1) + A}{A(A-1)} = 1 \]
This simplifies to:
\[ \frac{2A - 1}{A(A-1)} = 1 \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 2A - 1 = A^2 - A \]
Rearranging this equation results in:
\[ A^2 - 3A + 1 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ A = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ A = \frac{3 \pm \sqrt{9 - 4}}{2} \]
\[ A = \frac{3 \pm \sqrt{5}}{2} \]

### Step 7: Calculate C's time and work rate
Since C takes \( A - 1 \) hours, we can substitute \( A \):
\[ C = \frac{3 \pm \sqrt{5}}{2} - 1 = \frac{1 \pm \sqrt{5}}{2} \]

### Step 8: Determine C's work rate
The work rate of C is:
\[ \text{Work rate of C} = \frac{1}{C} = \frac{2}{1 \pm \sqrt{5}} \]

### Final Answer
To find how much work C does per hour, we can simplify this further, but the exact numerical value will depend on the positive root, as time cannot be negative.