Julie cashed a paycheck and repaid her brother $10 that she had borrowed from him. She then spent $30 on fuel for her car and half of the remaining money on a new tent for camping. She bought a pair of running shoes for $29.45 and had $17.75 left. How much did Julie receive when she cashed her paycheck?

Sometimes it's easier to work BACKWARDS, from the final remainder to the initial mount. 
In this case though, I'll go forward. 

Let paycheck amount be P
After repaying her brother $10 and buying fuel for $30, she had P - 10 - 30, or P - 40 remaining
After paying  of remainder for a tent, she had , remaining
After buying the shoes for $29.45, she had a FINAL remainder of 
As her FINAL remainder was $17.75, we get: 
                                                  
                                                   P - 40 = 94.4 ----- Cross-multiplying
                                         Paycheck, or P = 94.4 + 40 = $134.40

So, let's stsart from the end.


Before Julie bought a pait of running choes, she had 17.75 + 29.45 = 47.20 dollars.

This amount was the "second half" of her "remaining" money - so, the "remaining" money amount 
was 2*47.20 = 94.40 dollars .


Hence, her paycheck was $ 94.40  PLUS  $30  PLUS  $30,  i.e.  94.40 + 10 + 30 = 134.40 dollars.


ANSWER.  The paycheck was $134.40.