jane is 27

linda is 12

here's how:

let x = jane's age now.

let y = linda's age now.

7 years ago, jane was 4 times linda's age.

equation for that is:

x-7 = 4*(y-7)

simplify to get:

x-7 = 4y-28

solve for x to get:

x = 4y-21

when jane was 2 times linda's age, linda was 1/3 * jane's age.

this happened k years ago (we use letter k because we don't know how many years ago at this point in time).

equation for this is:

x-k = 2y

y-k = x/3

solve both these equations for k to get:

k = x-2y

k = y-x/3

since both equations are equal to k, you can set the expressions on the right of each equation equal to each other to get:

x-2y = y-x/3

multiply both sides of this equation by 3 to get:

3x-6y = 3y-x

add 6y to both sides of this equation and add x to both sides of this equation to get:

4x = 9y

divide both sides of this equation by 4 to get:

x = 9y/4

you now have 2 equations for x.

you have:

x = 4y-21

x = 9y/4

since both equations are equal to x, you can set the expressions on the right of each equation equal to each other to get:

4y-21 = 9y/4

multiply both sides of this equation by 4 to get:

16y-84 = 9y

subtract 9y from both sides of this equation and add 84 to both sides of this equation to get:

7y = 84

divide both sides of this equation by 7 to get:

y = 12

you should be able to use any of the equations to solve for x to get x = 27.

the one i used was the original one of:

x = 4y-21

this yielded x = 27

x = 9y/4 yielded x = 27 also.

you have x = 27 and y = 12

that's your answer.

to confirm, do the following from the original problem statement of:

when i was twice as old as you are now said Jane you were a third as old as i am now. Linda thought for a moment thats right she agreed and seven years ago you were 4 times as old as i was.

jane is 27

linda is 12.

7 years ago jane was 20 and linda was 5 which made jane 4 times as old as linda.

jane was twice as old as linda is now when jane was 24.

that was 3 years ago.

3 years ago, linda was 9 and and jane was 24 which made linda 1/3 as old as jane.

all portions of the problem statement check out so the answer is good.

CORRECTION TO PREVIOUS STATEMENT

3 years ago jane was 24 which is 2 times as old as linda is now (12).

3 years ago linda was 9 which is 1/3 times as old as jane is now (27).

answer is correct but last statement might have been confusing and so was corrected.