Troy’s monthly allowance is $42 more than Earl's. Earl spends $54 more than Troy every month. Earl 's savings is 1/2 of Troy's saving. If Troy spends 3/7 of his allowance every month, what is his allowance for the entire year?
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Let’s define a few symbols. Use E and T to identify Earl and Troy.

Use A, X and S for monthly allowance, expenditure and savings.

For example: T(S) means Troy’s savings, and T(S)=T(A)-T(X) shows how these savings are related to his monthly allowance and expenditure.

T(A)=E(A)+42, E(X)=T(X)+54, E(S)=½T(S), T(X)=(3/7)T(A).

From these, E(A)=T(A)-42, T(S)=2E(S), T(S)=T(A)-(3/7)T(A)=(4/7)T(A).

Also, T(S)=2E(S)=2(E(A)-E(X))=

2(T(A)-42-T(X)-54)=2(T(A)-T(X)-96)=2(T(S)-96).

So, T(S)=2(T(S)-96)=2T(S)-192.

We can solve this to find Troy’s monthly savings:

2T(S)-192=T(S), so T(S)=$192.

So 192=(4/7)T(A), T(A)=(7/4)192=7×48=$336.

This is Troy’s monthly allowance, so annually this is 12×336=$4032.

(If necessary, we can find the other figures, too:

E(A)=$294, T(X)=$144, E(X)=$198, T(S)=$192, E(S)=$96.)

 

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