John is 3 times as old as peter .in 6 years time john will be twice as old as peter will be . Determine their present ages
J = Present John's age
P = Present Peter's age
John is 3 times as old as peter mean:
J = 3 P
After 6 years John will be J + 6 yrs old
Peter will be P + 6 yrs old
In 6 years time john will be twice as old as peter will be mean:
( J + 6 ) / ( P + 6 ) = 2
Replace J = 3 P in this equation.
( 3 P + 6 ) / ( P + 6 ) = 2
Multiply both sides by P + 6
3 P + 6 = 2 ∙ ( P + 6 )
3 P + 6 = 2 ∙ P + 2 ∙ 6
3 P + 6 = 2 P + 12
Subtract 2 P to both sides
3 P - 2 P + 6 = 2 P - 2 P + 12
P + 6 = 12
Subtract 6 to both sides
P + 6 - 6 = 12 - 6
P = 6 yrs old
J = 3 P = 3 ∙ 6 = 18 yrs old
Proof:
J / P = 18 / 6 = 3
( J + 6 ) / ( P + 6 ) = ( 18 + 6 ) / ( 6 + 6 ) = 24 / 12 = 2
To determine their present ages, let's assign variables to their ages.
Let's say that John's age is "J" and Peter's age is "P".
According to the problem, John is 3 times as old as Peter, so we can write the equation: J = 3P.
In 6 years, John will be J+6 years old, and Peter will be P+6 years old. The problem also states that in 6 years, John will be twice as old as Peter will be, so we can write the equation: J+6 = 2(P+6).
Now we can solve the system of equations to find the values of J and P.
Substitute the value of J from the first equation into the second equation:
3P + 6 = 2(P + 6).
Distribute 2 to P + 6:
3P + 6 = 2P + 12.
Combine like terms:
3P - 2P = 12 - 6
P = 6.
Now, substitute the value of P into the first equation to find John's age:
J = 3(6)
J = 18.
Therefore, John is 18 years old and Peter is 6 years old.