# Jill is 6 years younger than Bill. Twelve years ago, Bill was 2 times older than Jill. How old are both of them now?

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To solve this problem, we'll translate the words into equations, then solve the equations.

Let *j* represent Jill's current age, and let *b* represent Bill's current age.

1.) "Jill is 6 years less than Bill"

`\implies j = b - 6`

2.) "Twelve years ago, Bill was 2 times older than Jill"

`\implies b - 12 = 2(j - 12)`

We will plug equation (1) into (2):

`\implies b - 12 = 2((b-6) - 12)`

We will now solve for b:

`\implies b - 12 = 2(b - 6) - 24`

`\implies b - 12 = 2b - 12 - 24`

`\implies b = 2b - 24`

`\implies b - 2b = -24`

`\implies b = 24`

Now we know that Bill is 24 years old. We can take this result, and plug it back into (1):

`\implies j = 24 - 6 = 18`

We now have our answer: **Bill is 24 years old and Jill is 18 years old.**

I arrived at the same answer without algebra. If Bill was 7 when Jill was 1, then when Jill was 6 Bill would have been 12, which is the only time he could have been twice Jill's age. Add 12 to each figure and you get 18 for Jill and 24 for Bill.

*Jill is six years younger than Bill*

Let Jill be x years old

Bills age = x+6

*12 years ago, Bill was 2 times older than bil*l

Age of Bill 12 years ago = x+6-12 = x-6

Age of Jill 12 years ago = x-12

As bills age was 2 times that of Jill therefore

x-6 = 2*(x-12)

x-6 = 2x-24

x-2x = -24+6

-x = -18

x = 18

Age of Jill now = 18 years

Age of Bill now = 18+6 = 24 years

*The age of Jill and Bill now is 18 and 24 years respectively*