8 years ago Rama's mother was five times older than her daughter. After 8 years the mother will be twice older than her daughter . Find the present age of Rama?

Answer

To find the present age of Rama, we can set up the problem using algebra based on the information given in the question.

### Step-by-Step Solution:

1. **Define Variables:**
Let \( X \) be the present age of Rama and \( Y \) be the present age of Rama's mother.

2. **Set Up the First Equation (from 8 years ago):**
Eight years ago, Rama's mother was five times older than her daughter. This can be expressed as:
\[ Y - 8 = 5(X - 8) \]
Simplifying this gives:
\[ Y - 8 = 5X - 40 \]
Rearranging it, we have:
\[ Y - 5X = -32 \quad \text{(Equation 1)} \]

3. **Set Up the Second Equation (from 8 years in the future):**
Eight years from now, Rama's mother will be twice as old as her daughter. This can be expressed as:
\[ Y + 8 = 2(X + 8) \]
Simplifying this gives:
\[ Y + 8 = 2X + 16 \]
Rearranging it, we have:
\[ Y - 2X = 8 \quad \text{(Equation 2)} \]

4. **Solve the System of Equations:**
Now we have two equations:
- Equation 1: \( Y - 5X = -32 \)
- Equation 2: \( Y - 2X = 8 \)

We can eliminate \( Y \) by subtracting Equation 2 from Equation 1:
\[ (Y - 5X) - (Y - 2X) = -32 - 8 \]
This simplifies to:
\[ -5X + 2X = -40 \]
\[ -3X = -40 \]
Dividing both sides by -3 gives:
\[ X = \frac{40}{3} \]

5. **Calculate the Present Age of Rama:**
Therefore, the present age of Rama is:
\[ X = \frac{40}{3} \approx 13.33 \text{ years} \]

### Final Answer:
The present age of Rama is \( \frac{40}{3} \) years, or approximately 13 years and 4 months.