Mrs. Mehra has two sons, one being exactly one year older than the other. At present, her age is equal to the sum of squares of ages of her sons. If 4 years hence her age becomes five times the age of the elder son then find the present ages of her sons.

Answer

To solve the problem step by step, let's denote the ages of Mrs. Mehra's sons and her age mathematically.

### Step 1: Define Variables
Let the age of the younger son be \( x \) years. Then, the age of the elder son will be \( x + 1 \) years.

### Step 2: Express Mrs. Mehra's Age
According to the problem, Mrs. Mehra's age is equal to the sum of the squares of her sons' ages. Therefore, we can express her age \( Y \) as:
\[ Y = x^2 + (x + 1)^2 \]

### Step 3: Expand the Expression
Now, we will expand the expression for Mrs. Mehra's age:
\[ Y = x^2 + (x^2 + 2x + 1) = 2x^2 + 2x + 1 \]

### Step 4: Set Up the Future Age Condition
The problem states that in 4 years, Mrs. Mehra's age will be five times the age of the elder son. In 4 years, Mrs. Mehra's age will be:
\[ Y + 4 = (2x^2 + 2x + 1) + 4 = 2x^2 + 2x + 5 \]
The age of the elder son in 4 years will be:
\[ (x + 1) + 4 = x + 5 \]
According to the problem, we have:
\[ 2x^2 + 2x + 5 = 5(x + 5) \]

### Step 5: Simplify the Equation
Now, we will simplify the equation:
\[ 2x^2 + 2x + 5 = 5x + 25 \]
Rearranging gives:
\[ 2x^2 + 2x + 5 - 5x - 25 = 0 \]
This simplifies to:
\[ 2x^2 - 3x - 20 = 0 \]

### Step 6: Factor the Quadratic Equation
Next, we will factor the quadratic equation:
\[ 2x^2 - 3x - 20 = 0 \]
We can factor this as:
\[ (2x + 5)(x - 4) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
1. \( 2x + 5 = 0 \) → \( x = -\frac{5}{2} \) (not valid since age cannot be negative)
2. \( x - 4 = 0 \) → \( x = 4 \)

### Step 8: Find the Present Ages of the Sons
Now that we have \( x = 4 \):
- The age of the younger son is \( 4 \) years.
- The age of the elder son is \( x + 1 = 4 + 1 = 5 \) years.

### Final Answer
The present ages of Mrs. Mehra's sons are:
- Younger son: **4 years**
- Elder son: **5 years**