SOLUTION: Hi Sarah spent 3/8 of her allowance on food and 1/3 of the remainder on clothes. Then she shares the rest of her allowance equally with her sisters such that each of them received

Hi
Sarah spent 3/8 of her allowance on food and 1/3 of the remainder on clothes. Then she shares the rest of her allowance equally with her sisters such that each of them received 1/12 of her total allowance. 

How many sisters does she have.
Given that they each received $208 dollars, how much does Sarah spend on clothes. 

After spending  of her allowance on food, Sarah had 
After spending  of the remainder on clothes, she had 
Sharing the remainder,  with her sisters, with each sister receiving  of her total allowance,
number of sisters remainder was shared with
Fraction of allowance each sister received: 
Amount of allowance each sister received: $208
Fraction of allowance spent on clothes: 
Amount of allowance spent on clothes: Let this be C
We then get the following PROPORTION: 
                                    ÷
                              
                                          2C = 5(208) ----- Cross-multiplying
         Amount spent on clothes, or 

There are more than 1 answer, because we are not given the number of sisters.

Starts with T.
Spends (3/8)T on food.  Has (5/8)T left.
Spends (1/3)(5/8)T = (5/24)T on clothes. Has (5/8)T-(5/24)T = (5/12)T left.
Shares equally with N sisters.
Since it says she shares equally with N sisters, that doesn't mean she gives
it all to them. She gives each of the siblings, including herself, (5/12)T/(N+1),
which is 208.

So we have the Diophantine equation:









Here is the way I was taught to solve first degree Diophantine equations.

Write each number in terms of its nearest multiple of the smallest coefficient
in absolute value. So write each 2496 as 2495+1 (because 2495 is the nearest
multiple of 5.




Divide through by 5



Isolate the fraction terms:



The left side is an integer, so the right side is also an integer.
Let that integer be A.




, the number of sisters

Substitute in











She spent  on clothes

A   Number of sisters  Spent on clothes
          5A-1              520A
--------------------------------
1           4                520
2           9               1040
3          14               1560
4          19               2080

There are multiple answers.

Edwin

Starts with T.
Spends (3/8)T on food.  Has (5/8)T left.
Spends (1/3)(5/8)T = (5/24)T on clothes. Has (5/8)T-(5/24)T = (5/12)T left.
Shares equally with N sisters.
Since it says she shares equally with sisters, that doesn't mean she gives
it all to them. 
She gives each of the siblings, including herself, (1/12)T,
which is 208 dollars.


So, to compute the number of sisters, including Sarah, we should divide (5/12)T by (1/12)T.

Obviously, the quotient is 5.  So, the number of sisters (including Sarah) is 5.


Also, from the analysis above,  (1/12)T = 208;  hence,  T = 208*12 = 2496 dollars.


The amount, which Sarah spent on clothes is (5/24)*2496 = 520 dollars.


ANSWER.  Sarah has 5-1 = 4 sisters.

         The amount Sarah spent on clothes was $520.

Sorry, after scrolling up to the posted problem I forgot to put down all the
data. In this case the "1/12 of her total allowance."  Oh well, I'll chalk it up
to being an 87-year old codger! J

Edwin