The sum of the ages of Aryabhatta and Shridhar is 45 years. Five years ago the product of their ages was 4 times the Aryabhatta's age at that tim. The present ages of Aryabhatta and Shridhar respectively are:

Options

A. 25 and 20
B. 35 and 10
C. 36 and 9
D. 40 and 5

Answer

To solve the problem step by step, we will set up equations based on the information provided about the ages of Aryabhatta and Shridhar.

### Step 1: Define the Variables
Let:
- \( x \) = Aryabhatta's current age
- \( y \) = Shridhar's current age

### Step 2: Set Up the First Equation
According to the problem, the sum of their ages is 45 years. Therefore, we can write:
\[ x + y = 45 \]

### Step 3: Set Up the Second Equation
The problem states that five years ago, the product of their ages was 4 times Aryabhatta's age at that time. Five years ago, Aryabhatta's age would have been \( x - 5 \) and Shridhar's age would have been \( y - 5 \). Thus, we can write the second equation as:
\[ (x - 5)(y - 5) = 4(x - 5) \]

### Step 4: Simplify the Second Equation
Expanding the left side of the equation:
\[ xy - 5x - 5y + 25 = 4x - 20 \]
Now, rearranging gives:
\[ xy - 5x - 5y + 25 - 4x + 20 = 0 \]
This simplifies to:
\[ xy - 9x - 5y + 45 = 0 \]

### Step 5: Substitute \( y \) from the First Equation
From the first equation \( x + y = 45 \), we can express \( y \) in terms of \( x \):
\[ y = 45 - x \]
Now substitute \( y \) into the second equation:
\[ x(45 - x) - 9x - 5(45 - x) + 45 = 0 \]

### Step 6: Expand and Simplify
Expanding the equation:
\[ 45x - x^2 - 9x - 225 + 5x + 45 = 0 \]
Combining like terms:
\[ -x^2 + 41x - 180 = 0 \]
Multiplying through by -1 to make the leading coefficient positive:
\[ x^2 - 41x + 180 = 0 \]

### Step 7: Factor the Quadratic Equation
We need to factor the quadratic equation:
\[ (x - 36)(x - 5) = 0 \]
Setting each factor to zero gives:
\[ x - 36 = 0 \quad \text{or} \quad x - 5 = 0 \]
Thus, \( x = 36 \) or \( x = 5 \).

### Step 8: Determine the Ages
If \( x = 36 \):
\[ y = 45 - 36 = 9 \]
If \( x = 5 \):
\[ y = 45 - 5 = 40 \]
However, since the ages must be realistic in the context of the problem, we take \( x = 36 \) and \( y = 9 \).

### Conclusion
The present ages of Aryabhatta and Shridhar are:
- Aryabhatta: 36 years
- Shridhar: 9 years