SOLUTION: Tom, Alice, Helen and Sally had $430 altogether. Alice and Helen had the same amount of money. Alice had 2/3 of the amount Tom had left after he had spent $86. Sally was given $16

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Question 1202336: Tom, Alice, Helen and Sally had $430 altogether. Alice and Helen had the same amount of money. Alice had 2/3 of the amount Tom had left after he had spent $86. Sally was given $16 and she had as much as Alice. How much money did Sally have at first?
Found 4 solutions by math_tutor2020, josgarithmetic, ikleyn, MathTherapy:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Answer: $64


Work Shown:

H = Helen's amount
A = Alice's amount
T = Tom's amount
S = Sally's amount

Let's list what info we're given
Fact 1Tom, Alice, Helen and Sally had $430 altogether.
Fact 2Alice and Helen had the same amount of money.
Fact 3Alice had 2/3 of the amount Tom had left after he had spent $86.
Fact 4Sally was given $16 and she had as much as Alice.

Those facts lead to these four equations
H+A+T+S = 430 due to fact 1
A = H because of fact 2
A = (2/3)*(T-86) because of fact 3
S+16 = A because of fact 4

I'll label those equation (1) through equation (4).

Solve equation (3) for T
A = (2/3)*(T-86)
3A = 2(T-86)
3A = 2T-172
3A+172 = 2T
2T = 3A+172
T = (3A+172)/2
T = 1.5A+86

Then we'll plug in A = S+16
T = 1.5A+86
T = 1.5(S+16)+86
T = 1.5S+24+86
T = 1.5S+110

Let's move to equation (1)
H+A+T+S = 430
A+A+T+S = 430 ... plug in A = H
2A+T+S = 430
2(S+16)+T+S = 430 ... plug in A = S+16
2S+32+T+S = 430
3S+32+T = 430
3S+32+1.5S+110 = 430 ... plug in T = 1.5s+110
4.5S+142 = 430
4.5S = 430-142
4.5S = 288
S = 288/4.5
S = 64
Answer by josgarithmetic(39615)   (Show Source): You can put this solution on YOUR website!
               ORIGINAL    
Tom               t
Alice             x
Helen             x
Sally             s
SUM              430


Tom spent $86.


Sally was given $16.


----



-

-
substitute into SUM equation.






------------No. This line is wrong. Should be .


Question asks effectively for s.





(Possibly made arithmetic mistake - not rechecked)
Answer by ikleyn(52772)   (Show Source): You can put this solution on YOUR website!
.

            I made a necessary correction in the solution by @josgarithmetic. 


               ORIGINAL    
Tom               t
Alice             x
Helen             x
Sally             s
SUM              430


Tom spent $86.


Sally was given $16.


----



-

-
substitute into SUM equation.






        <<<---===   it is the correction point


Question asks effectively for s.
,
        ANSWER


Answer by MathTherapy(10551)   (Show Source): You can put this solution on YOUR website!
Tom, Alice, Helen and Sally had $430 altogether. Alice and Helen had the same amount of money. Alice had 2/3 of the amount Tom had left after he had spent $86. Sally was given $16 and she had as much as Alice. How much money did Sally have at first?

Let amount Sally had, be S    
Then Alice had: S + 16     
Since Helen and Alice had the same amount, then Helen’s was also S + 16
Since total amount was $430, Tom then had: 430 - (S + S + 16 + S + 16) = 430 - (3S + 32) = 430 - 3S - 32 = 398 - 3S 
After spending $86, Tom then had remaining amount of, 398 - 3S - 86 = 312 - 3S
Since Alice had 2/3 of Tom’s, after Tom spent $86, we then get:  
                                                                S + 16 = 2(104 - S)
                                                                S + 16 = 208 - 2S
                                                                S + 2S = 208 - 16
                                                                    3S = 192
                                       Initial amount Sally had, or

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