Solution for "How Old Are They?" 

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Question:  A year from now, David's age will be twice Jane's age.  Two years ago, David's age was three times Jane's age.  How old are David and Jane right now?


Let \(J=\mbox{Jane's current age in years} \).
Let \(D=\mbox{David's current age in years} \).
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Next year, David's age will be \(D+1 \), and Jane's age will be \(J+1 \).  So, if we say that next year David's age will be twice Jane's age, we can write that as 
\[D+1=2(J+1)\\
\mbox{or}\\
D=2J+1\]

Two years ago, David's age was \(D-2 \), and Jane's age was \(J-2 \).  So, if we say that two years ago David's age was three times Jane's age, we can write that as
\[D-2=3(J-2)\\
\mbox{    or    }\\
​D=3J-4 \]
Okay, now we have two equations with two unknowns:
1.  \(D=2J+1 \)
2.  \(D=3J-4 \)
We can subtract Equation (1) from Equation (2) so that the \(D \)'s cancel, and then we have \(0=J-5 \) or \(J=5 \).  

Plugging \(J=5 \) back into Equation (1) gives \(D=2(5)+1=11 \)

Therefore, Jane's current age is 5, and David's current age is 11.  
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