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 two sons as follows:

- Let the age of the younger son (S1) be \( x \) years.
- Therefore, the age of the elder son (S2) will be \( x + 1 \) years (since he is one year older).

### Step 1: Set up the equation for Mrs. Mehra's current age
According to the problem, Mrs. Mehra's current age (M) is equal to the sum of the squares of her sons' ages. Therefore, we can write the equation as:

\[ M = x^2 + (x + 1)^2 \]

### Step 2: Expand the equation
Now, let's expand the right side of the equation:

\[ M = x^2 + (x^2 + 2x + 1) = 2x^2 + 2x + 1 \]

### Step 3: Set up the equation for ages 4 years hence
The problem also states that in 4 years, Mrs. Mehra's age will be five times the age of the elder son. In 4 years, the ages will be:

- Mrs. Mehra's age: \( M + 4 \)
- Elder son's age: \( (x + 1) + 4 = x + 5 \)

Thus, we can set up the equation:

\[ M + 4 = 5(x + 5) \]

### Step 4: Substitute \( M \) into the equation
Now, substitute \( M \) from the previous equation into this new equation:

\[ (2x^2 + 2x + 1) + 4 = 5(x + 5) \]

This simplifies to:

\[ 2x^2 + 2x + 5 = 5x + 25 \]

### Step 5: Rearrange the equation
Now, let's rearrange the equation to bring all terms to one side:

\[ 2x^2 + 2x + 5 - 5x - 25 = 0 \]

This simplifies to:

\[ 2x^2 - 3x - 20 = 0 \]

### Step 6: Factor the quadratic equation
Next, we will factor the quadratic equation \( 2x^2 - 3x - 20 = 0 \). To do this, we look for two numbers that multiply to \( 2 \times -20 = -40 \) and add to \( -3 \). The numbers are \( -8 \) and \( 5 \).

Thus, we can rewrite the equation as:

\[ 2x^2 - 8x + 5x - 20 = 0 \]

Now, factor by grouping:

\[ 2x(x - 4) + 5(x - 4) = 0 \]

This gives us:

\[ (2x + 5)(x - 4) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:

1. \( 2x + 5 = 0 \) → \( x = -\frac{5}{2} \) (not possible since age cannot be negative)
2. \( x - 4 = 0 \) → \( x = 4 \)

### Step 8: Find the ages of the sons
Now that we have \( x = 4 \), we can find the ages of the sons:

- Younger son (S1) = \( x = 4 \) years
- Elder son (S2) = \( x + 1 = 5 \) years

### Conclusion
Thus, the present ages of Mrs. Mehra's sons are:

- Younger son: 4 years
- Elder son: 5 years