Question 5300
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/n}}}.
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"
{{{ (C+3)*(n-2) = 48 + 22}}} where {{{C=48/n}}}


Substitute for the C in this equation:
{{{ (48/n + 3) (n-2) = 70 }}}


Find the Least Common Denominator for the binomial above:
{{{  ((48+ 3n)/n )* (n-2) = 70}}}
{{{  ((3n+ 48)/n )* (n-2) = 70}}}
{{{  ((3n^2 -6n + 48n -96)/n ) = 70}}}


Multiply both sides by the denominator "n":
{{{ 3n^2 + 42n - 96 = 70n }}}


Subtract 70n from each side of the equation to set it equal to zero:

{{{3n^2 + 42n - 70n - 96 = 70n-70n}}}
{{{3n^2 - 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^2 - 4ac}}} you get 1936 and {{{sqrt (1936) = 44}}}.  So, solve it however you can . . .


If you factor it, this comes out to 

{{{3n^2 - 28n - 96= 0}}}

{{{(3n+8)(n-12) = 0}}}
{{{n= -8/3}}} {{{n= 12}}}


n cannot be a negative fraction, so 
n= 12 mugs were purchased.
n-2 = 10 mugs were sold.

{{{48/n= 48/12 = 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