Four years ago Jane was twice as old as Sam. Four years from now Sam will be three-fourths of Jane's age. How old is Jane now?

4 years ago Jane was 8, Sam was 4. In 4 years from now, Jane will be 16 and Sam will be 12.

So Jane is 12 years old now.

Let's turn this into an algebra problem

We'll use J for Jane's age now, and S for Sam's age now, and make an equation which shows the relationship four years ago.

J - 4 = 2(S - 4) and we'll work out the brackets and move the unknowns to the same side

J - 4 = 2S - 8

J - 2S = - 8 + 4

J - 2S = - 4

Now we'll do another equation to show the relationship in four years time

3/4(J + 4) = S + 4 --- and we'll multiply both sides by 4 to get rid of that fraction

3(J + 4) = 4(S + 4)

3J + 12 = 4S + 16

3J - 4S = 16 - 12

3J - 4S = 4

Now we need to make one of the unknown the same value so we can subtract the two equations.... if we multiply the top one by -3 we get

-3J +6S = 12, and if we add that to the second equation we will get a new equation with no J value

3J - 4S = 4

-3J +6S = 12

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0J + 2S = 16

Therefore S = 8, and we can now substitute 8 for S in any of the other equations, and check the answer we get for J

J - 2(8) = - 4

J - 16 = - 4

J = - 4 + 16

J = 12

Now, if we are right then substituting these values not the second equation should give a correct sum.

3J - 4S = 4

3(12) - 4(8) = 4

36 - 32 = 4

Which is correct,.... so we know that the calculations are right.

To put that whole puzzle into real number then we look at he first statement

Four years ago Jane (who was then 12-4) was 8. The was twice as old as Sam (who was then 8 - 4) who was 4.

Four years from now Sam (will be 8 + 4) at 12, will be three-quarters of Jane's (who will be 12 + 4) age of 16. Correct answer is 12.