Question 1209504
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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 {{{highlight(cross(cost))}}} <U>costs</U> $3.30 more than one diary. 
How much money did they have altogether. 
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<pre>
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  

    {{{(9/5)*(2x)}}} = {{{(18/5)x}}}.


From it, we should take  4/9  of  {{{(18/5)x}}}  to calculate the part of Grace's money which left.  So,

    Grace's money left =  {{{(4/9)*(18/5)x}}} = {{{((4*2)/5)x}}} = {{{(8/5)x}}}


    +----------------------------------------------------+
    |   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  <U>money left</U>  is

    x + 3.30 = {{{(8/5)x}}}.    ( ! ! ! - 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*3.30)/3}}} = 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  {{{(4/3)*26.40}}} = 35.20 dollars.



Grace spent 2*5.50 = 11 dollars.  It is 5/9 of her starting money.

Hence, Grace's starting money was  {{{(9/5)*11}}} = 19.80 dollars.



Thus the starting money of two girls, altogether, was  35.20 + 19.80 = 55 dollars.


<U>ANSWER</U>.  Initially, the girls had  55 dollars, altogether.
</pre>

Solved.


This problem is nice.  It teaches to think and forces to think.


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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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<U>Comment from student</U>: 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



<U>My response</U>:  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:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- Bad/wrong problem formulations obtain their true rank, which they deserve;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- obscure formulations/tasks become clear, 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- and all reasonable tasks become solved and explained.