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Kelly had 3 times as much money as Andy. After giving Andy $450, Kelly had twice as much money as he did. (a) How much money did Andy have in the end? (b) How much money did Kelly have in the beginning?

Boy oh boy -- working on this grade 6 problem (embarrassingly enough I have a master in physics and am struggling with the logic!)

My logical solution (that doesn't work and I don't know why it doesn't!):

Let $$K_0$$ = Amount of money Kelly had initially $$A_0$$ = Amount of money Andy had initially $$K_{new}$$ = Amount of money Kelly had after giving money to Andy $$A_{new}$$ = Amount of money Andy had after receiving money from Kelly

We know Kelly had 3 times as much as Andy so:

$$3 K_0 = A_0$$

We then know Kelly gave $450 to Andy so:

$$K_{new} = K_0 - 450$$ $$A_{new} = A_0 + 450$$

Then we know with this new balance Kelly has twice as much as Andy so:

$$2 K_{new} = A_{new}$$

When solving this system of equations, I get negative numbers and the math doesn't work. I looked at the solutions and they implement some model method (which I have never heard of). I am more curious to know where the flaw in my logic is with the equations I put forth.

A seemingly easy problem for my son has drove me to the brink! Please help :)

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    $\begingroup$ kelly had 3xAndy, $K_0=3A_0$ $\endgroup$ Commented Oct 9, 2022 at 18:26
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    $\begingroup$ And similarly $K_\text{new} = 2A_\text{new}$. $\endgroup$
    – Brian Tung
    Commented Oct 9, 2022 at 18:28
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    $\begingroup$ oh dear god..... $\endgroup$
    – sci-guy
    Commented Oct 9, 2022 at 18:30
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    $\begingroup$ Happens to everyone. Or so I assert :-). $\endgroup$
    – copper.hat
    Commented Oct 9, 2022 at 18:37
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    $\begingroup$ This is a grade 6 "Singapore" level for what it's worth. $\endgroup$
    – sci-guy
    Commented Oct 9, 2022 at 18:47

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