Question 1209491
<pre>
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

{{{4h=expr(3/8)(x-90)}}}

Then she had 
{{{(x-90)-expr(3/8)(x-90)}}} left. That's
{{{(x-90)(1-3/8)}}}
{{{(x-90)(5/8)}}}
{{{expr(5/8)(x-90)}}} left

Then she gave 2/5 of that to her mother.  That's 
{{{(2/5)*expr(5/8)(x-90)}}} or
{{{expr(2/8)(x-90)}}} that she gave to her mother,
so she had
{{{expr(5/8)(x-90)}}}{{{""-""}}}{{{expr(2/8)(x-90)}}} or

{{{expr(3/8)(x-90)}}} left

Assuming that she saved ALL the rest,

{{{expr(3/8)(x-90)=45}}}

Multiply both sides by 8

{{{3(x-90)=360}}}
{{{x-90=120}}}
{{{x=210}}} <-- 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

{{{4h=expr(3/8)(x-90)}}}
{{{4h=expr(3/8)(210-90)}}}
{{{4h=expr(3/8)}}}(120}}}
{{{4h=45}}}
{{{h=11.25}}}

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</pre>