Question

A girl is twice as old as her sister. Four years hence, the product of thier ages (in years) will be 160. Find their present ages.

Answer 1

Let the age of her sister = x
Then the girl's age = 2x

After 4 years,
Age of girl = 2x+4
Age of her sister = x+4

Products is 160
Or (2x+4)×(x+4) = 160

Or 2x^2 + 12x + 16 = 160

Or 2x^2 + 12x - 144 = 0

Or x^2 + 6x - 72 = 0

Or x^2 + 12x - 6x - 72 = 0

Or (x+12)(x-6)=0

Or x = -12 or 6

Since age can not be negative, x = 6
2x = 2×6 = 12 years

So age of her sister is 6 years and age of the girl is 12 years.

Answer 2

Let the present age of the girl is 2x years and age of her sister is x years




After 4 years,
Age of the girl = ( 2x + 4 ) years
Age of her sister = ( x + 4 ) years,








After 4 years, product of their ages = 160



=> ( 2x + 4 ) ( x + 4 ) = 160


=> 2x² + 8x + 4x + 16 = 160


=> 2x² + 12x + 16 = 160


=> x² + 6x + 8 = 80


=> x² + 6x + 8 - 80 = 0


=> x² + 6x - 72 = 0


=> x² + ( 12 - 6 ) x - 72 = 0


=> x² + 12x - 6x - 72 = 0


=> x( x + 12 ) - 6( x + 12 ) = 0


=> ( x + 12 ) ( x - 6 ) = 0



Hence, x = -12 or x = 6



Taking positive value because age cant be negative,

Hence, x = 6







Then,





Present age of the girl = 2x = 2( 6 ) = 12 years

$.$

Present age of her Sister = x = 6 years