SOLUTION: When John was as old as Paul is now, the sum of their ages was 51. When Paul will be as old as John is now, the sum of their ages will be 103. John is older than Paul by how many

Question 1127332: When John was as old as Paul is now, the sum of their ages was 51. When Paul will be as old as John is now, the sum of
their ages will be 103. John is older than Paul by how many years?
Answer by FrankM(1040) (Show Source): You can put this solution on YOUR website!
"When John was as old as Paul is now"
The key thing is to understand the set-up.
Say John is 30, and Paul is 20, the phrase above is the difference in their ages. J-P, or 10 years ago. . And both of them 'were' J-P (or here, 10) years younger. The 30/20 was an example.
J+P-2(J-P)= 51
Next is J-P years from now. So we add.
J+P+2(J-P)= 103

Next, simplify the first eq.
J+P-2J+2P=51
-J+3P = 51
And the second
J+P+2(J-P)= 103
3J-P = 103 (next, triple the first equation to add it)
-3J+9P=153
8P=256
P = 32 and J = 45
13 years ago (J-P), P was 19, J was 32, these add to 51.
In 13 years, P will be 45, J, 58, adding to 103.

You can also see that the way the question is set up, the sum of their ages changes (103-51) years, or by 52. From X years ago to X years hence, we can see that their age difference must be 52/4 or 13. 26 years pass from the point in past to point in future, times the 2 people. Once you see the 13 is true, the set-up can use 13 instead of J-P.

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