Direction: The question given below is followed by three statements. Study the question and the statements. Identify which option is necessary to answer the question.
Three workers A, B and C complete a given piece of work within different time spans, while working individually. What is the ratio of efficiency of C to that of B?
I. A takes `100%` more time than C to complete the given piece of work.
II. B completes the given piece of work in 8 days.
III. B takes 2 days less than A to complete the given piece of work.

Options

A. Question cannot be answered even with the information in all three statements.
B. All of the three
C. Only II and III
D. II and either only I or only III

Answer

To solve the problem of finding the ratio of efficiency of worker C to worker B, we will analyze the statements provided step by step.

### Step 1: Analyze Statement I
Statement I states that "A takes `100%` more time than C to complete the given piece of work."

- Let's assume C takes `x` days to complete the work.
- If A takes `100%` more time than C, then A takes `x + 100% of x = x + x = 2x` days.

### Step 2: Analyze Statement II
Statement II states that "B completes the given piece of work in 8 days."

- This gives us the time taken by B directly: B = 8 days.

### Step 3: Analyze Statement III
Statement III states that "B takes 2 days less than A to complete the given piece of work."

- Since we know B takes 8 days, we can express this in terms of A:
\[ B = A - 2 \implies 8 = A - 2 \implies A = 8 + 2 = 10 \text{ days} \]

### Step 4: Relate A and C
From Statement I, we know:
- A takes `2x` days and we found A takes 10 days.
- Therefore, we can set up the equation:
\[ 2x = 10 \implies x = 5 \text{ days} \]
So, C takes 5 days to complete the work.

### Step 5: Calculate Efficiencies
Efficiency is inversely proportional to the time taken to complete the work. Therefore, we can express the efficiencies as:
- Efficiency of B = \( \frac{1}{8} \)
- Efficiency of C = \( \frac{1}{5} \)

### Step 6: Find the Ratio of Efficiencies
To find the ratio of efficiency of C to that of B:
\[ \text{Ratio of efficiency of C to B} = \frac{\text{Efficiency of C}}{\text{Efficiency of B}} = \frac{\frac{1}{5}}{\frac{1}{8}} = \frac{8}{5} \]

### Conclusion
Thus, the ratio of efficiency of C to that of B is \( 8:5 \).