The last sentence says

    "After buying the hairpins, she gave 2/5 of her money she had left to her mother and saved $45".


It means that $45 is 3/5 of her money after buying the hairpins.

Hence, her money after buying the hairpins was   = 75 dollars.



Now we analyze next statement back from the end, which is

    "she spent 3/8 of her remaining money on 4 identical hairpins". 


OK. She spent 3/8 of her remaining money, which means that 5/8 of remaining money was exactly 75 dollars,
    identified one step above.

    Hence, the remaining money at this context is   = 120 dollars.

    And now we determine that 3/8 of this $120 she spent for four hairpins

         = 45 dollars.



Thus, 4 hairpins cost $45 - - - hence, each hairpin costs   = 11.25  dollars.


At this point, the problem is solved completely, and we have the


ANSWER.  Each hairpin costs $11.25.



There is no need to analyze further, since we just answered the problem's question.

She solved it backwards.  I'll solve it forward, which will probably require
a little more work, but it'll be more straight forward.

However, there is a slight flaw in the problem. We are not told that when she
saved $45, that this $45 was ALL she had left.  She could have put $45 in her
savings account and still had some left over.  But I will assume that she saved
ALL that was left.  For otherwise, the problem would not be solvable.

Suppose she had $x at the beginning and each hairpin cost $h.

After spending $90 on a blouse and skirt, she had $(x-90) left.

So then she paid 4h for hair pins and that was equal to 3/8 of $(x-90). So



Then she had 
 left. That's


 left

Then she gave 2/5 of that to her mother.  That's 
 or
 that she gave to her mother,
so she had
 or

 left

Assuming that she saved ALL the rest,



Multiply both sides by 8



 <-- so she had $210 to start with, which was not asked for, but
                  could have been. And which you would need to check the problem.

Substituting in



(120}}}



So each hairpin cost $11.25.  <--the answer!

Checking:

Rachel had $210 to begin with 

Rachel spent $90 on a blouse and shoes. 

So she had $120 left.

After that she spent 3/8 of her remaining money on 4 identical hairpins. 

Each hairpin cost $11.25 so 4 of them cost $45. So she had $120-$45 = $75 left.

After buying the hairpins, she gave 2/5 of her money she had left to her mother

So she gave 2/5 of $75 or $30 to her mother.

So she had $75 - $30 = $45 left

And since she saved ALL of it, our work is correct.

Edwin