Question 1209504: Hi
Jamie spent 3/4 of her money on 3 books of the same price. Grace spent 5/9 of her money on 2 diaries. They were each left with an equal amount of money. One book cost $3.30 more than one diary. How much money did they have altogether.
Found 3 solutions by CPhill, greenestamps, ikleyn:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **1. Set up the equations:**
* Let 'x' be the cost of one diary.
* Then, the cost of one book is 'x + $3.30'.
* Jamie spent 3/4 of her money on 3 books.
* Cost of 3 books: 3 * (x + $3.30)
* Fraction of money spent by Jamie: 3/4
* Grace spent 5/9 of her money on 2 diaries.
* Cost of 2 diaries: 2 * x
* Fraction of money spent by Grace: 5/9
* Since they had the same amount of money left, the fractions of money they spent must be equal:
* (Cost of 3 books) / (Fraction spent by Jamie) = (Cost of 2 diaries) / (Fraction spent by Grace)
* [3 * (x + $3.30)] / (3/4) = (2 * x) / (5/9)
**2. Solve the equation:**
* Simplify the equation:
* 4 * (3x + $9.90) = (18/5) * 2x
* 12x + $39.60 = (36/5) * x
* Isolate 'x':
* 12x - (36/5) * x = -$39.60
* (60x - 36x) / 5 = -$39.60
* 24x = -$39.60 * 5
* 24x = -$198
* x = -$198 / 24
* x = -$8.25
* **Cost of one diary: -$8.25**
* **Cost of one book:** -$8.25 + $3.30 = -$4.95
**3. Calculate the total money they had:**
* Since the cost of a diary is negative, there seems to be an error in the problem statement.
* Assuming the problem is solvable, you would use the cost of a diary (x) and the fraction of money spent by either Jamie or Grace to calculate their initial amount of money.
* Then, you would add the amounts of money Jamie and Grace initially had to find the total.
**Therefore, based on the given information, there appears to be an error in the problem, as the calculated cost of a diary is negative.**
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
Another bizarre and incorrect AI solution from that pseudo-tutor....
"Since they had the same amount of money left, the fractions of money they spent must be equal"
That makes no logical sense. And, besides, the statement of the problem tells us the fractions of their money that each spent, and they are not equal.
Following is a logical and correct solution from a human tutor....
Let J and G respectively represent the amounts of money Jamie and Grace started with.
Let x be the cost of each diary; then the cost of each book was x+$3.30.
Jamie spent 3/4 of her money, so she finished with 1/4 of what she started with.
Grace spent 5/9 of her money, so she finished with 4/9 of what she started with.
They finished with the same amounts of money:
[1] 
Jamie spent 3/4 of her money on 3 books:
[2] 
Grace spent 5/9 of her money on 2 diaries:
[3] 
The amount of money Jamie had left was  of what she spent:
[4] 
The amount of money Grace had left was  of what she spent:
[5] 
The amounts they ended up with were the same:
[6] 
The cost of each diary was x=$5.50; the cost of each book was x+$3.30 = $8.80.
Jamie ended up with x+$3.30 = $8.80; that was 1/4 of what she started with, so she started with 4($8.80) = $35.20.
Grace ended up with was (8/5)x = $8.80; that was 4/9 of what she started with, so she started with (9/4)($8.80) = 9($2.20) = $19.80.
The total the two of them started with together was $35.20+$19.80 = $55.
ANSWER: $55
Summary....
Jamie started with $35.20 and spent 3/4 of it ($26.40) on 3 books costing $8.80 each, finishing with 1/4 of $35.20, or $8.80.
Grace started with $19.80 and spent 5/9 of it ($11.00) on 2 diaries costing $5.50 each, finishing with 4/9 of $19.80, or $8.80.
The two of them finished with equal amounts.
Answer by ikleyn(52835)  (Show Source):
You can put this solution on YOUR website! .
Jamie spent 3/4 of her money on 3 books of the same price.
Grace spent 5/9 of her money on 2 diaries.
They were each left with an equal amount of money.
One book  costs $3.30 more than one diary.
How much money did they have altogether.
~~~~~~~~~~~~~~~~~~~~~~~
Let x be the price of one diary, in dollars.
Then the price of one book is (x+3.30) dollars, according to the problem.
Jamie spent 3/4 of her money and bought 3 books.
Hence, one book costs 1/4 of Jamie's money.
But 1/4 of Jamie's money is exactly the amount which left in Jamie's possession..
So, we conclude that Jamie left (x+3.30) dollars.
+-----------------------------------------------------------------------------+
| OK, it is fine. So, (x+3.30) dollars is the left side of the equation, |
| which we are going to built to solve the problem. |
+-----------------------------------------------------------------------------+
Now let's calculate on the Grace side.
Grace spent 2x dollars, which is 5/9 of her starting money. Hence, Grace's starting money was
 =  .
From it, we should take 4/9 of to calculate the part of Grace's money which left. So,
Grace's money left =  =  = 
+----------------------------------------------------+
| Thus, Grace's money left is (8/5)*x dollars. |
| It is the right side of our equation for |
| money left. |
+----------------------------------------------------+
And now the equation for money left is
x + 3.30 = . ( ! ! ! - here three exclamation signs express my delight)
To solve, multiply by 5 both sides of this equation
5x + 5*3.30 = 8x
5*3.30 = 8x - 5x
5*3.30 = 3x
x =  = 5*1.10 = 5.50.
So, one diary costs 5.50 dollars and one book costs $3.30 more, or 5.50+3.30 = 8.80 dollars.
Now we are in the finish line.
Jamie spent 3*8.80 = 26.40 dollars. It is 3/4 of her starting money.
Hence, Jamie's starting money was  = 35.20 dollars.
Grace spent 2*5.50 = 11 dollars. It is 5/9 of her starting money.
Hence, Grace's starting money was  = 19.80 dollars.
Thus the starting money of two girls, altogether, was 35.20 + 19.80 = 55 dollars.
ANSWER. Initially, the girls had 55 dollars, altogether.
Solved.
This problem is nice. It teaches to think and forces to think.
------------------------
Perhaps, some tutors will come, offering other (more shorter or more "iron language" solutions).
I wrote my solution trying to make it accessible for as young students
as possible, when a student is able to understand it.
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Ignore the post by @CPhill, since it exploits wrong logic.
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Comment from student: Hi ikleyn I am glad you enjoyed this problem. I honestly thought I was going to get a roasting
for posting a badly worded question. This textbook my son uses for extra work is very good,
however some of the questions...the wording makes your head spin. 😀 you are a GREAT TUTOR
My response: I am glad to see your adequate reaction, which rarely happens at this forum.
Can you, PLEASE, post me the name of the textbook your son uses for extra work.
Thank you in advance.
. . . . . . . . . . . . . . . .
Please, do not afraid my criticism. After it, all things become better:
- Bad/wrong problem formulations obtain their true rank, which they deserve;
- obscure formulations/tasks become clear,
- and all reasonable tasks become solved and explained.
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