The problem of ages
You Had an age we'll call x is today HAS an age we'll call y.
I HAVE twice the age you had when I was your current age y (twice as x), ie I I have 2x years.
SO:
You We had x and now there is y.
I HAD y and now I have 2x.
Therefore we have to:
yx = 2xy
2y = 3x
x = (2/3) * y
So, replacing the value of x, we have:
You THINGS (2/3) * y and now there is y.I HAD y and now I have (4/3) * y.
Now pay attention to the second sentence:
WHEN YOU HAVE MY AGE, THE SUM OF OUR AGES WILL BE 45 YEARS OLD.
You have y, and to be my age, which is (4/3) * y, you must add your age y with more (1/3) * y.
Adding y + (1/3) * y you will be my age, meaning you will have (4/3) * y.
How we add (1/3) * y at your age, we should add to mine too, namely:
Now I have (4/3)*y + (1/3) * y, soon I have (5/3) * y.
The sum of our ages must equal 45 years:
(4/3) * y + (5/3) * y = 45
(9/3) * y = 45
3y = 45
y = 15
At first we found that x = (2/3) * y, so x = (2/3) * 15, so x = 10.
FINALLY: WHAT ARE OUR AGES ???
As we said at the beginning, your current age is y, THAT IS, 15 YEARS.
AND MY AGE IS 2x, Ie 2.10, which is equal to 20 YEARS.
SO THE AGES ARE 20 AND 15 YEARS!!!
Warning 1: Every day we get emails from users saying that this answer is wrong, because adding the ages we don't get 45. However, note that the statement does not say that the current sum of ages is 45, but that "When you are my age, the sum of our ages will be 45", that is, when 15 is 20, 20 is already 25 (20 + 25 = 45). Notice 2: Some people get confused by saying that there is another possible answer for the current ages, which would be 9 and 18 years. But this answer is not valid because it does not satisfy the first sentence of the statement, which says "I'm twice as old as you were when I was your age"because when 18 was 9, 9 was zero. And we know that 9 is not double zero. 
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