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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?

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    $\begingroup$ 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. $\endgroup$ Nov 21, 2021 at 6:02
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    $\begingroup$ The strategy is, 1) identify the unknowns, 2) give them names, 3) write down equations relating the unknowns, 4) solve those equations. $\endgroup$ Nov 21, 2021 at 6:03
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    $\begingroup$ $D =$ 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, ... $\endgroup$ Nov 21, 2021 at 6:06
  • $\begingroup$ 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 $T=3(2K-C)$ $\endgroup$ Nov 21, 2021 at 7:13

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

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