# How can I simplify then solve this wordy age problem?

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?

• You start by defining variables, $T$ for Tom's age now and $K$ for Katy's age now. The last sentence gives $T+K=26$. You have to read the sentence before carefully to get an equation out of it. Alternatively, you know that $T$ is a multiple of $3$. There are not many of those below $26$, so you can just try them by plugging in. Commented Nov 21, 2021 at 6:02
• The strategy is, 1) identify the unknowns, 2) give them names, 3) write down equations relating the unknowns, 4) solve those equations. Commented Nov 21, 2021 at 6:03
• $D =$ current day. $T+K = 26$. At time $D_1,$ Kay was $K/2$ years old, so $D - K/2 = D_1$. At $D_2$, Tom was $4 \times K/2 = 2K$, so $D_2 = D - (T - 2K).$ And on, and on, ... Commented Nov 21, 2021 at 6:06
• The key to simplifying this problem is in the realization that there is a constant difference in the two ages: $T-K=C$ This should lead you to $T=3(2K-C)$ Commented Nov 21, 2021 at 7:13

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
-> $$X = 3\cdot[Y-(X-2Y)]$$
-> $$4X=9Y$$