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 the time. 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, let's denote the present ages of Aryabhatta and Shridhar as \( A \) and \( S \) respectively.

### Step 1: Set up the equations
From the problem, we know:
1. The sum of their ages is 45 years:
\[ A + S = 45 \quad \text{(Equation 1)} \]

2. Five years ago, the product of their ages was 4 times Aryabhatta's age at that time:
\[ (A - 5)(S - 5) = 4(A - 5) \quad \text{(Equation 2)} \]

### Step 2: Simplify Equation 2
Expanding Equation 2:
\[ (A - 5)(S - 5) = AS - 5A - 5S + 25 \]
And simplifying the right side:
\[ 4(A - 5) = 4A - 20 \]
So, we can set the two sides equal:
\[ AS - 5A - 5S + 25 = 4A - 20 \]

### Step 3: Rearranging the equation
Rearranging gives us:
\[ AS - 5A - 5S + 25 - 4A + 20 = 0 \]
This simplifies to:
\[ AS - 9A - 5S + 45 = 0 \quad \text{(Equation 3)} \]

### Step 4: Substitute \( S \) from Equation 1 into Equation 3
From Equation 1, we can express \( S \) in terms of \( A \):
\[ S = 45 - A \]
Now substitute \( S \) into Equation 3:
\[ A(45 - A) - 9A - 5(45 - A) + 45 = 0 \]
Expanding this:
\[ 45A - A^2 - 9A - 225 + 5A + 45 = 0 \]
Combining like terms:
\[ -A^2 + 41A - 180 = 0 \]
Multiplying through by -1 gives:
\[ A^2 - 41A + 180 = 0 \]

### Step 5: Solve the quadratic equation
Now we can solve this quadratic equation using the quadratic formula:
\[ A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1 \), \( b = -41 \), and \( c = 180 \):
\[ A = \frac{41 \pm \sqrt{(-41)^2 - 4 \cdot 1 \cdot 180}}{2 \cdot 1} \]
Calculating the discriminant:
\[ A = \frac{41 \pm \sqrt{1681 - 720}}{2} \]
\[ A = \frac{41 \pm \sqrt{961}}{2} \]
\[ A = \frac{41 \pm 31}{2} \]
Calculating the two possible values for \( A \):
1. \( A = \frac{72}{2} = 36 \)
2. \( A = \frac{10}{2} = 5 \)

### Step 6: Find corresponding \( S \)
Using \( A = 36 \):
\[ S = 45 - 36 = 9 \]
Using \( A = 5 \):
\[ S = 45 - 5 = 40 \]

### Conclusion
Thus, the present ages of Aryabhatta and Shridhar are:
- Aryabhatta: 36 years
- Shridhar: 9 years

However, the ages must be reasonable, so we check:
- If Aryabhatta is 36, then Shridhar is 9, which makes sense.
- If Aryabhatta is 5, then Shridhar is 40, which is also valid.

### Final Answer
The present ages of Aryabhatta and Shridhar respectively are:
- Aryabhatta: 36 years
- Shridhar: 9 years