Question 1204068
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x = her paycheck amount
x-10 = what she has left over after repaying her brother
x-10-30 = x-40 = left over amount after buying fuel


(1/2)(x-40) = 0.5x-20 = amount she has left after buying the tent


0.5x-20-29.45 = 0.5x-49.45 = amount she has left after buying the shoes


Set this equal to the stated value 17.75 so we can solve for x.
0.5x-49.45 = 17.75
0.5x = 17.75+49.45
0.5x = 67.20
2*0.5x = 2*67.20
x = <font color=red>$134.40</font> is the answer.


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Checking the answer:


start = $134.40
repays brother = $134.40 - $10 = $124.40 left over
buys fuel = $124.40 - $30 = $94.40 left over
buys tent = (1/2)*($94.40) = $47.20 left over
buys shoes = $47.20 - $29.45 = $17.75 is what she'll have left over after purchasing all the items mentioned, and repaying her brother.
Answer is confirmed.



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Another approach


She ended up with $17.75
Add on the price of the shoes to rewind the clock so she gets that money back.
17.75+29.45 = 47.20
Before buying the shoes, she would have $47.20


Double this to get $94.40 when we rewind the clock further back. This is the amount she would have just before buying the tent.
Then add on $30 to rewind further to get $124.40 to represent what she would have just before buying fuel.


Lastly, she'll take the $10 from her brother when rewinding the clock all the way back to the start point.
$124.40+$10 = <font color=red>$134.40</font>
Basically we follow the "check" section shown above, but do so in reverse. 
Any time we have a subtraction, the inverse would be addition. The inverse of halving is doubling. 
This rewinding action is basically how we solve equations. 


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Answer: <font color=red size=4>$134.40</font>
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