How can I simplify then solve this wordy age problem?

Question

About a month ago, my high school had its first math team competition. In one of the events (Fr/So 8-person), the following question was asked:

"All ages in this problem are in whole number of years. Tom is now 3 times as old as Kay was when Tom was 4 times as old as Kay had been when Kay was 1/2 as old as Kay is now. If the sum of their present ages is 26 years, find the number of years in Tom's age now.

What would be a strategy that I could use to simplify it down into easier to understand equations, and then to solve it?

Answer 1

I think the main problem lies in this sentence:

Tom is now 3 times as old as Kay was when Tom was 4 times as old as Kay had been when Kay was 1/2 as old as Kay is now.

Let Tom be $X$ years old and Kay be $Y$ years old. Break down from back to front:

1. when Kay was 1/2 as old as Kay is now
   -> when Kay was $\frac Y2$ years old
   -> $\frac Y2$ years ago.

2. when Tom was 4 times as old as Kay had been when Kay was 1/2 as old as Kay is now.
   -> when Tom was 4 times as old as Kay had been $\frac Y2$ years ago.
   -> when Tom was $4 \cdot \frac Y2$ years old
   -> $X-2Y$ years ago

3. Tom is now 3 times as old as Kay was when Tom was 4 times as old as Kay had been when Kay was 1/2 as old as Kay is now.
   -> Tom is now 3 times as old as Kay was $X-2Y$ years ago
   -> $X = 3\cdot[Y-(X-2Y)]$
   -> $4X=9Y$