In eight years, Jack will be 2/3 as old as Tom. Four years ago he was 1/3 as old as Tom. Find their present ages.
In eight years, Jack will be 2/3 as old as Tom. Four years ago he was 1/3 as old as Tom. Let the age of Jack be J and that of Tom be T.
According to the data provided
J + 8 = (2/3)*(T + 8) and J - 4 = (1/3)*(T - 4)
=> 3J + 24 = 2T + 16 and 3J - 12 = T - 4
Subtracting the two equations gives
T + 20 = 36
=> T = 16
and J = 8
Presently Jack is 8 and Tom is 16.
Let Jack's present age be x and Tom's present age be y
If we write the condition given in the question to equations,
x+8 = 2/3(y+8)
x-4 = 1/3(y-4)
In the first equation,
3x+24 = 2y+16
=> y = 3/2x + 4
Then we can plug in the equation of y into the second equation
x-4 = 1/3(3/2x+4-4) = 1/2x
Therefore, solving the last equation, x-4 = 1/2x,
x = 8
y = 3/2*8 +4 = 16
Jack's present age is 8, Tom's present age is 16