Given below is a question followed by three statements . You have to study the question and the ststements and decide which of the statement is/are necessary to answer the question .
What is Arun's present age ?
I. Five years ago, Arun's age was double that of his son's age at that time .
II. Present ages of Arun and his son are in the ratio of `11:6` respectively .
III. Five years hence, the respective ratio of Arun's age and his son's age will become `12:7` .

Options

A. Only I and II
B. Only II and III
C. Only I and III
D. Any two of the three

Answer

To determine Arun's present age based on the given statements, let's analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** Five years ago, Arun's age was double that of his son's age at that time.

Let Arun's present age be \( A \) and his son's present age be \( S \).

Five years ago:
- Arun's age = \( A - 5 \)
- Son's age = \( S - 5 \)

According to Statement I:
\[ A - 5 = 2(S - 5) \]
Expanding this gives:
\[ A - 5 = 2S - 10 \]
Rearranging it:
\[ A = 2S - 5 \]
This equation relates Arun's age to his son's age but does not provide a specific value for Arun's age. Therefore, Statement I alone is **not sufficient**.

### Step 2: Analyze Statement II
**Statement II:** Present ages of Arun and his son are in the ratio of 11:6.

Let the present age of Arun be \( 11x \) and the present age of his son be \( 6x \).

This gives us:
- Arun's age = \( 11x \)
- Son's age = \( 6x \)

This statement provides a relationship between their ages but does not give a specific value for Arun's age. Therefore, Statement II alone is also **not sufficient**.

### Step 3: Analyze Statement III
**Statement III:** Five years hence, the respective ratio of Arun's age and his son's age will become 12:7.

Five years hence:
- Arun's age = \( A + 5 \)
- Son's age = \( S + 5 \)

According to Statement III:
\[ \frac{A + 5}{S + 5} = \frac{12}{7} \]
Cross-multiplying gives:
\[ 7(A + 5) = 12(S + 5) \]
Expanding this:
\[ 7A + 35 = 12S + 60 \]
Rearranging it:
\[ 7A - 12S = 25 \]
This equation relates Arun's age to his son's age.

### Step 4: Combine Statements II and III
Now we have two equations:
1. From Statement II: \( A = 11x \) and \( S = 6x \)
2. From Statement III: \( 7A - 12S = 25 \)

Substituting \( A \) and \( S \) from Statement II into the equation from Statement III:
\[ 7(11x) - 12(6x) = 25 \]
This simplifies to:
\[ 77x - 72x = 25 \]
\[ 5x = 25 \]
\[ x = 5 \]

Now substituting back to find Arun's age:
\[ A = 11x = 11 \times 5 = 55 \]

### Conclusion
Arun's present age is **55 years**.

### Summary of Necessary Statements
- **Statement I** is not sufficient alone.
- **Statement II** is not sufficient alone.
- **Statement III** is not sufficient alone.
- **Statements II and III together are sufficient** to determine Arun's present age.