John & Julia's Ages: Solving the Mystery

[Jeebus]

Question posed:
Julia is as old as John will be when Julia is twice as old as John was when Julia's age was half the sum of their present ages.

John is as old as Julia was when John was half the age he will be 10
years from now.

How old are John and Julia?
______________________________
First I let x = John's present age and y = Julia's present age

x + 10 = John's age in 10 years

x + 10
------ = 1/2 of John's age in 10 years
2

Now, it would be helpful to know when John was (x + 10)/2. Let's say a person is 56 years old and if I want to know when was I 20, I just subtract 56 - 20 and find that I was 20, 36 years ago.
x + 10
So, x - ------ will tell us how many years ago John was (x + 10)/2.
2

How old was Julie then? Well, if I want to know how old my 42-year-old
brother was 36 years ago, I just subtract 42 - 36 and learn that he
was 6.
x + 10
So, Julie's age must have been y - (x - -------). But, if that is what
2
John's age is right now, we have:

x + 10
x = y - ( x - ------ )
2

I went through this same reasoning process with the rest of the
information in the problem and came up with another equation in two
variables. Is that right?

 

[arcnets]

Originally posted by Jeebus
we have:

x + 10
x = y - ( x - ------ )
2

Yes, that is right. But since this system doesn't support blanks so well, you should rather type:
x = y - (x - (x+10)/2).

That's the correct equation for statement (2).
As you correctly stated, statement (1) will give another equation in x and y, and this system of 2 equations in 2 unknowns should have a unique solution.

 

[M98Ranger]

Yes, your reasoning and equations are correct. To solve for x and y, we can set the two equations equal to each other and solve for one variable in terms of the other. This would give us a value for one of the ages, which we can then substitute into either equation to solve for the other age. This process may be a bit tedious, but it is an effective way to solve for both John and Julia's ages in this mystery.

 

[M98Ranger]

Yes, you are correct. Your reasoning process is correct and you have set up the correct equations to solve for John and Julia's ages. Now, we can solve for x and y by setting the two equations equal to each other and solving for one variable. Substituting the value of x into either equation will then give us the value of the other variable. This will give us the ages of John and Julia. Great job in breaking down the problem and setting up the equations!