Question

Anjali has two sons, one being exactly 2 years older than the other. At present, her age is equal to the sum of the squares of the ages of her sons. If 6 years hence her age will be eight times the present age of the elder son, find their present ages.​

Answer 1

$\bold{\huge{\underline{ Solution }}}$

Given :-

• Anjali has two sons
• One son is exactly 2 years older than the other
• At present, Anjali age is equal to the sum of the squares of the ages of her sons
• 6 years later, Anjali age will be 8 times the present age of the elder son

To Find :-

• We have to find the present ages of the Anjali and his sons

Let's Begin :-

Let the present age of the younger son be x

Elder son is exactly 2 years older than younger son

Therefore,

• The age of elder son is x + 2

According to the question,

The age of Anjali is equal to the sum of the squares of the ages of her sons

That is

$\sf{ = x^{2} + ( x + 2)^{2}}$

$\sf{ = x^{2} + x^{2} + 4 + 2x }$

$\sf{ = 2x^{2} + 4x + 4 }$

Now,

We have to find the age of mother and her sons after 6 years

After 6 years

The age of the younger son

$\sf{ = (x + 6)\: years }$

Therefore ,

The age of elder son

$\sf{ = (x + 2) + 6 }$

$\sf{ = x + 2 + 6 }$

$\sf{ = x + 8 \: years }$

Here, It is given that,

• After 6 years the age of Anjali will be 8 times his elder son's age

That is,

$\sf{ = 2x^{2} + 4x + 4 + 6 = 8( x + 8)}$

$\sf{ = 2(x^{2} + 2x + 2 + 3) = 8(x + 2)}$

$\sf{ = x^{2} + 2x + 2 + 3 = 4(x + 2)}$

$\sf{ = x^{2} + 2x + 5 = 4x + 8}$

$\sf{ = x^{2} + 2x - 4x = 8 - 5}$

$\sf{ = x^{2} - 2x = 3 }$

$\sf{ = x^{2} - 2x - 3 = 0 }$

By factorization method

$\sf{ =(x + 1)( x -3 )}$

$\sf{ = ( x = -1 ) or ( x = 3 )}$

• Age of the person can never be negative

Thus, The present age of the younger son is 3 years

Therefore,

Age of the elder brother

$\sf{ = 3 + 2}$

$\bold{ = 5 \: years}$

Present age of Anjali

$\sf{ = 2(3)^{2} + 4(3) + 4 }$

$\sf{ = 2(9) + 12 + 4 }$

$\sf{ = 18 + 16 }$

$\bold{ = 34\: years }$

Hence, The present age of Anjali and his two sons is 34 years, 5 years and 3 years .