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

Algebra ->  Percentage-and-ratio-word-problems -> 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       Log On


   

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) About Me  (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) About Me  (Show Source):
You can put this solution on YOUR website!
               ORIGINAL    
Tom               t
Alice             x
Helen             x
Sally             s
SUM              430


Tom spent $86.
x=%282%2F3%29%28t-86%29

Sally was given $16.
s%2B16=x

----
3x=2%28t-86%29
3x%2F2=t-86
t=3x%2F2%2B86
-
s=x-16
-
substitute into SUM equation.
highlight_green%283x%2F2%2B86%2B2x%2B%28x-16%29=430%29

3x%2F2%2B3x-16%2B86=430
3x%2F2%2B3x=430%2B16-86
3x%2F2%2B3x=360
3x%2B6x=720
x=cross%28720%2F8%29------------No. This line is wrong. Should be x=720%2F9.
cross%28x=90%29

Question asks effectively for s.
s=x-16
s=cross%2890-16%29
highlight%28cross%28s=74%29%29


(Possibly made arithmetic mistake - not rechecked)

Answer by ikleyn(52771) About Me  (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.
x=%282%2F3%29%28t-86%29

Sally was given $16.
s%2B16=x

----
3x=2%28t-86%29
3x%2F2=t-86
t=3x%2F2%2B86
-
s=x-16
-
substitute into SUM equation.
highlight_green%283x%2F2%2B86%2B2x%2B%28x-16%29=430%29

3x%2F2%2B3x-16%2B86=430
3x%2F2%2B3x=430%2B16-86
3x%2F2%2B3x=360
3x%2B6x=720
x=720%2F9         <<<---===   it is the correction point
highlight%28x=80%29

Question asks effectively for s.
s=x-16, s=80-16
highlight%28highlight%28s=64%29%29         ANSWER


Answer by MathTherapy(10551) About Me  (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