SOLUTION: A retailer spent $48 to purchase a number of mugs. Two of them were broken in the store, but by selling each of the remaining mugs for $3 above the original cost per mug, she made

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Question 5300: A retailer spent $48 to purchase a number of mugs. Two of them were broken in the store, but by selling each of the remaining mugs for $3 above the original cost per mug, she made a total profit of $22.
If the price for n mugs is $48, how can I express the cost per mug?
The number of mugs that are available for sale?
Rewrite the equation from above in the standard form of a quadratic equation?
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
This is an interesting problem. Let's start by letting
n= number of mugs purchased by the retailer
n-2 = number of mugs sold (two were broken and not able to be sold!)
C = cost per mug paid by the retailer
C + 3 = retail selling price of the mugs (for which retailer gets a $3 profit)

To answer your other questions:
If the price for n mugs is $48, then the cost per mug is $48%2Fn.
The number of mugs available for sale = n-2.

The equation is based upon the revenue from the sale of the mugs:
"Selling price per mug TIMES number of mugs sold = Revenue = Cost + Profit"
+%28C%2B3%29%2A%28n-2%29+=+48+%2B+22 where C=48%2Fn

Substitute for the C in this equation:
+%2848%2Fn+%2B+3%29+%28n-2%29+=+70+

Find the Least Common Denominator for the binomial above:
++%28%2848%2B+3n%29%2Fn+%29%2A+%28n-2%29+=+70
++%28%283n%2B+48%29%2Fn+%29%2A+%28n-2%29+=+70
++%28%283n%5E2+-6n+%2B+48n+-96%29%2Fn+%29+=+70

Multiply both sides by the denominator "n":
+3n%5E2+%2B+42n+-+96+=+70n+

Subtract 70n from each side of the equation to set it equal to zero:
3n%5E2+%2B+42n+-+70n+-+96+=+70n-70n
3n%5E2+-+28n+-+96=+0

Now this is a GREAT place to use Ichudov's new process for solving quadratic equations, but unfortunately I haven't learned it yet. Maybe the Maestro will step in and add his touch to this solution. Anyway, I DO know that this equation DOES factor, since if you calculate b%5E2+-+4ac you get 1936 and sqrt+%281936%29+=+44. So, solve it however you can . . .

If you factor it, this comes out to
3n%5E2+-+28n+-+96=+0
%283n%2B8%29%28n-12%29+=+0
n=+-8%2F3 n=+12

n cannot be a negative fraction, so
n= 12 mugs were purchased.
n-2 = 10 mugs were sold.
48%2Fn=+48%2F12+=+4 $4 = cost to buy each mug
Add a $3 profit to each mug = $7 for each of the 10 mugs that were sold.

Check:
12 mugs were purchased at $4 each for a total of $48.
10 mugs were sold at $7 each for a total of $70. That gives a profit of $22.

R^2 at SCC