John is FIVE years older than Peter.Twice the product of their ages FIVE years ago is 100 more than product of their present ages.Find the present ages of John and Peter.

Answer 1

J = John´s present age

P = Peter´s present age

John is five years older than Peter mean:

J = P + 5

Five years ago John was J - 5 yrs old , Peter was P - 5 yrs old.

Twice the product of their ages five years ago is 100 more than product of their present ages mean:

2 ∙ ( J - 5 ) ∙ ( P - 5 ) = 100 + J ∙ P

Replace J = P + 5 in this equation.

2 ∙ ( P + 5 - 5 ) ∙ ( P - 5 ) = 100 + ( P + 5 ) ∙ P

2 ∙ P ∙ ( P - 5 ) = 100 + ( P + 5 ) ∙ P

2 ∙ P ∙ P - 2 ∙ P ∙ 5 = 100 + P ∙ P + P ∙ 5

2 P² - 10 P = 100 + P² + 5 P

Subtract P² to both sides

P² - 10 P = 100 + 5 P

Subtract ( 100 + 5 P ) to both sides

P² - 10 P - ( 100 + 5 P ) = 0

P² - 10 P - 100 - 5 P = 0

P² - 15 P - 100 = 0

The solutions are P = - 5 and P = 20

The present ages can't be negative so:

P = 20

J = P + 5 = 20 + 5 = 25

John´s present age = 25

Peter´s present age = 20

Proof:

Five years ago John was 25 - 5 = 20 yrs old , Peter was 20 - 5 = 15 yrs old.

2 ∙ ( J - 5 ) ∙ ( P - 5 ) = 100 + J ∙ P

2 ∙ 20 ∙ 15 = 100 + 25 ∙ 20

600 = 600

Answer 2

Peter is X yrs. old.
John is x+5 yrs. old.

Five yrs. ago:
2(x-5)(x+5-5) = x(x+5) + 100.
2(x-5)x = x^2+5x+100,
2x^2-10x = x^2+5x+100,
x^2-15x = 100,
x^2-15x-100 = 0, -100 = 5*(-20). sum = -15 = B.
(x+5)(x-20) = 0.
x+5 = 0, X = -5.
x-20 = 0, X = 20 yrs. = Peters age.
x+5 = 20+5 = 25 yrs. = John's age.