The given question is followed by three statements. You have to study the question and all the three statements to decide which of the statements is/are necessary to answer the given question.
What is the present age of Tanya ?
I. The ratio between the present ages of Tanya and her brother Rahul is 3 : 4 respectively.
II. After 5 years, the ratio between the ages of Tanya and Rahul will be 4 : 5.
III. Rahul is 5 years older than Tanya.

Options

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

Answer

To determine Tanya's present age based on the given statements, we will analyze each statement step by step.

### Step 1: Understanding the Statements
1. **Statement I**: The ratio between the present ages of Tanya and her brother Rahul is 3:4.
- Let Tanya's age be \(3x\) and Rahul's age be \(4x\).

2. **Statement II**: After 5 years, the ratio between the ages of Tanya and Rahul will be 4:5.
- After 5 years, Tanya's age will be \(3x + 5\) and Rahul's age will be \(4x + 5\).
- The ratio can be set up as:
\[ \frac{3x + 5}{4x + 5} = \frac{4}{5} \]

3. **Statement III**: Rahul is 5 years older than Tanya.
- This can be expressed as:
\[ R = T + 5 \quad \text{(where \(R\) is Rahul's age and \(T\) is Tanya's age)} \]
- Substituting the expressions from Statement I:
\[ 4x = 3x + 5 \implies x = 5 \]

### Step 2: Solving the Equations
#### Using Statement I and II:
1. From Statement I, we have:
- Tanya's age \(T = 3x\)
- Rahul's age \(R = 4x\)

2. Substitute \(x\) from Statement II:
\[ \frac{3x + 5}{4x + 5} = \frac{4}{5} \]
Cross-multiplying gives:
\[ 5(3x + 5) = 4(4x + 5) \]
Expanding both sides:
\[ 15x + 25 = 16x + 20 \]
Rearranging gives:
\[ 15x - 16x = 20 - 25 \implies -x = -5 \implies x = 5 \]

3. Now substituting \(x\) back to find Tanya's age:
\[ T = 3x = 3(5) = 15 \]

### Step 3: Conclusion
- Tanya's present age is **15 years**.