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Nguyen has opened a new bake shop in town selling pies. Nguyen sells each pie for $ 9 $9 and has spent $ 1 , 311 $1,311 on rent and maintenance this month. How many pies did Nguyen sell if he made $ 399 $399 profit this month?
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Let x x be the number of pies Nguyen sold.
The revenue from selling x x pies is the product of the price per pie and the number of pies:
Revenue = 9 x =9x
The cost of selling x x pies is the sum of the fixed costs (rent and maintenance) and the cost per pie (which we don’t know):
Cost = 1 , 311 + c x =1,311+cx
where c c is the cost per pie.
The profit for selling x x pies is:
Profit = 9 x ( 1 , 311 + c x ) =9x-(1,311+cx)
We know that Nguyen made a profit of $ 399 $399, so we can set up the equation:
9 x ( 1 , 311 + c x ) = 399 9x-(1,311+cx)=399
Solving for c c using the information given in the problem:
399 = 9 x ( 1 , 311 + c x ) 399=9x-(1,311+cx)
399 = 9 x 1 , 311 c x 399=9x-1,311-cx
c x = 9 x 1 , 710 cx=9x-1,710
c = 9 x 1 , 710 x c=(9x-1,710)/(x)
Plugging this expression for c c back into the equation:
9 x ( 1 , 311 + 9 x 1 , 710 x ) = 399 9x-(1,311+(9x-1,710)/(x))=399
Multiplying both sides by x x to eliminate the fraction:
9 x 2 1 , 311 x + 9 x 1 , 710 = 399 x 9x^(2)-1,311 x+9x-1,710=399 x
Simplifying:
9 x 2 703 x 1 , 710 = 0 9x^(2)-703 x-1,710=0
We can solve for x x using the quadratic formula:
x = ( 703 ) ± ( 703 ) 2 4 ( 9 ) ( 1710 ) 2 ( 9 ) x=(-(-703)+-sqrt((-703)^(2)-4(9)(-1710)))/(2(9))
Simplifying:
x = 703 ± 497 , 209 + 61 , 560 18 x=(703+-sqrt(497,209+61,560))/(18)
x = 703 ± 558 , 769 18 x=(703+-sqrt(558,769))/(18)
x = 703 ± 747 18 x=(703+-747)/(18)
The positive root gives the number of pies that Nguyen sold:
x = 703 + 747 18 = 1450 18 80.56 x=(703+747)/(18)=(1450)/(18)~~80.56
Rounding down since Nguyen can’t sell a fraction of a pie, he sold 80 80 pies.
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