Each of the following question is followed by three statements I, II and III. You have to study the question and the statements and decide which of the statements is/are necessary to answer the question.
What is the ratio of the presentage of Ajay and Akshay?
I. 6 years ago, ratio of age of Ajay to Akshay is 3 : 4.
II. After 2 years, the younger one's age will be 85% of the older.
III. The sum of their ages is three less than twice the age of older.

Options

A. Only statements II and III together are sufficient.
B. Any two of them is sufficient.
C. Only statements I and II together are sufficient.
D. All statements are not sufficient to answer.

Answer

To solve the problem of finding the ratio of the percentage of Ajay and Akshay's ages, we will analyze each statement provided and determine their relevance to answering the question.

### Step-by-Step Solution:

1. **Understanding the Question**:
We need to find the ratio of the ages of Ajay and Akshay in percentage terms. This means we need to express Ajay's age as a percentage of Akshay's age.

2. **Analyzing Statement I**:
- Statement I states that "6 years ago, the ratio of the age of Ajay to Akshay is 3:4."
- Let Ajay's age 6 years ago be \(3x\) and Akshay's age be \(4x\).
- Therefore, Ajay's current age is \(3x + 6\) and Akshay's current age is \(4x + 6\).
- This statement provides a relationship between their ages, but we cannot find the ratio of their current ages directly from this statement alone.

**Hint**: This statement gives historical ages but does not provide enough information to find the current ratio.

3. **Analyzing Statement II**:
- Statement II says, "After 2 years, the younger one's age will be 85% of the older."
- If we assume Ajay is the younger one, then after 2 years, Ajay's age will be \(3x + 8\) and Akshay's age will be \(4x + 8\).
- According to the statement, \(3x + 8 = 0.85(4x + 8)\).
- This equation can be solved to find the relationship between their ages and eventually the ratio.

**Hint**: This statement provides a direct relationship that can help us find the current ages.

4. **Analyzing Statement III**:
- Statement III states, "The sum of their ages is three less than twice the age of the older."
- The sum of their current ages is \((3x + 6) + (4x + 6) = 7x + 12\).
- The age of the older (Akshay) is \(4x + 6\), so twice that is \(2(4x + 6) = 8x + 12\).
- According to the statement, \(7x + 12 = (8x + 12) - 3\).
- This can be simplified to find a relationship, but it is not directly useful for finding the ratio.

**Hint**: This statement provides a condition but does not directly help in finding the ratio of their ages.

5. **Conclusion**:
- From our analysis, Statement I alone is not sufficient.
- Statement II is crucial as it gives a direct relationship that can be solved for the ages.
- Statement III provides additional information but is not necessary to find the ratio.
- Therefore, the combination of Statement I and Statement II is sufficient to find the ratio of their ages.

### Final Answer:
The necessary statements to answer the question are **Statements I and II together**.