In which year will I be twice your age?

This question, of course, depends on who you are, and who "I" is*, but we can work out a general case using algebra. First, though, I'll use myself and my friend John as:


An example
If we know the year of birth of any particular person we can work out their age by subtracting their birth year from the current year.

• TeaKay was born in 1982 and it's now 2009, so my age is      2009 - 1982 = 27      years old**.
• John was born in 1955, and it's still 2009, so his age is      2009 - 1955 = 54      years old.

Now, you can see already that John will become twice my age this year (2009), but how could we work it out if it wasn't so easy?

How would we work out how old each person is in any given year? Well, we'd use the same method- subtract their birth year from the year you're looking at. In general, we could say that:

• TeaKay is/was/will be      y - 1982      years old in the year y.
• John is/was/will be      y - 1955      years old in the year y.

To answer the question we want to find what year a must be for John's age to be twice TeaKay's age. So we can write:     John's age = 2 x TeaKay's age

Algebraically, that would look like this:

     y - 1955     =     2 x (y - 1982)

Now all we have to do is solve to find a value for a:

     y - 1955     =     2y - 3964               Expand the brackets

     y + 2009    =     2y                          Use the inverse to get the numbers on the same side of the '=' sign

            2009   =       y                          Use the inverse to get the letters on the same side of the '=' sign

So now we have proven that John will be twice as old as TeaKay in 2009.


A general case
We can go a step further and develop a formula for finding out when anyone will be twice as old as anyone else:

Call your people Person A and Person B. Person A is always the oldest of the two. Now call a the year that Person A was born in, and b the year that Person B was born in. In the year y:

• Person A will be      y - a      years old.
• Person B will be      y - b      years old.

So Person A will be twice as old as Person B when:

     y - a      =      2(y - b)

So now we can solve to find a value for y, given any years a and b:

     y - a      =      2y - 2b

  y - a + 2b =      2y

      -a + 2b =       y

And neaten up:

                y = 2b - a

So, in English, this formula says "to find the year in which Person A will be twice as old as Person B, double Person B's birth year and then subtract Person A's birth year."

Which is why mathematicians like algebra so much!

*Difficult to parse, I know, but grammatically sound in the sense that I mean it...
** Of course this assumes that your birthday has already happened- I won't actually be 27 until October 10th, but lets simplify in order to get the idea across.