Found 7 solutions by josgarithmetic, mananth, MathTherapy, greenestamps, Edwin McCravy, ikleyn, AnlytcPhil:
Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
T, the TOTAL allowance before any spending and sharing
----
Sarah spent 3/8 of her allowance on food and 1/3 of the remainder on clothes.
----
First remain and second, remain which is ,
before anything goes to her and her sisters.
If EACH of them is given of T, then this exactly allows for of these portions.
Those are 1 for Sarah and then her 4 sisters.
---
Given that they each received $208 dollars, how much does Sarah spend on clothes.
---
Plainly means .
Total initial allowance was then . Dollars.
;
-----------how much Sarah spent on clothes.
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
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.
Sarah spent 3/8 of her allowance on food and 1/3 of the remainder on clothes.
Let her allowance be x
Spent 3x/8 on food
Balance = x-3x/8 = 5x/8
She spent (1/3)*5x/8 on clothes = 5x/24
Now balance with her (5x/8)- (5x/24) = 5x/12
Let number of sisters be n
each sister got (1/12)x
Total given to sisters = n(1/12)x
Balance with her was 5x/12
5x/12 = n * (1/12)x
n = 5
There were 5 sisters
Each sister got (1/12) x
208 = (1/12)x
x = 2496 Total allowance
Money spent on clothes = 5x/24
= 5*2496/24
= 520
Money spent on clothes = 520
Check
Money spent on clothes = 520
Money spent on food = 3x/8 = 3*2496/8 = 936
Money given to sisters = 5*208 = 1040
520+936+1040 =2496 which was her allowance.
Answer by MathTherapy(10552) (Show Source): You can put this solution on YOUR website!
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:
รท
2C = 5(208) ----- Cross-multiplying
Amount spent on clothes, or
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
The response you have received so far agree on everything except the number of sisters.
Sarah has 5/12 of her allowance left after buying food and clothes. Since she SHARES that with her sisters with each of them (her sisters AND Sarah herself) getting 1/12 of her allowance, the number of portions is 5 -- one for Sarah and one for each of her FOUR sisters.
ANSWER (number of sister): 4
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!
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
Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website!
.
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.
~~~~~~~~~~~~~~~~~~
The treatment of the problem by Edwin is incorrect.
It is incorrect, because Edwin ignored (did not use)
the condition that every sibling got 1/12 of the Sarah's total allowance.
So, in Edwin' solution this last condition was, actually, ignored.
This brought Edwin to Diophantine equation, while the true logic is much simpler.
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.
Solved.
--------------------
Edwin's error is a typical error what mathematicians make when they solve simple problems:
they over-complicate them in their minds.
Answer by AnlytcPhil(1806) (Show Source): You can put this solution on YOUR website!
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
RELATED QUESTIONS