Algebraic problem: forming equation?

[buddyboole19]

1. A retailer spent $48 to purchase a number of special mugs. Two of them were broken in the store, but by selling each of the remaining mugs for $3 above the orignial cost per mug, she made a total profit of $22. Construct an equation that will allow us to solve for the number of mugs, denoted by n, that were originally purchased.How many mugs were purchased? What was the original cost per mug? What profit was made on each mug?


So first - my original cost per mug equation: C=48/n
Selling price of mug: S=(48/n)+3
number of mugs sold: n-2
number of mugs bought: n

my attempt at solving: (n-2) * ((48/n)+3) = 48 +22

my thought process here was i took the number of mugs able to be sold (n-2) multiplied by the selling price of each mug = ((48/n)+3) to get the total money made 48+22. Where 22 is the profit she made, and 48 was the original purchase price.

so first thing i did was get rid of the N in the denominator of selling price. multiplying (n-2) times n, and ((48/n)+3) times n = and then 70 times n

simplifying gets me (n^2-2n) * (48 +3n) = 70
multiplying with distributive property :
3n^3+42n^2-96n = 70 n
isolate 0 gets me

3n^3 +42n^2 -166n=0

factor out one n:

n(3n^2+42n-166)=0

now I am trying to factor my quadratic equation. But i just cant. nothing works.
Ive long since figured N to be 12, simply by trial an error. But i need to show equations. And i have no idea where my math went in error. Can anyone show me an equation to use, or where i went wrong or help me to get the proper answer. Thank you in advance

 

[berkeman]

Your original equation is correct: (n-2) * ((48/n)+3) = 48 +22

I think this is where the algebra error happens:

(n^2-2n) * (48 +3n) = 70

When you multiply both sides of your original equation by n, you should have gotten:

(n-2) * ((48)+3n) = 70n

Does that get you a cleaner quadratic equation?

BTW, when I did this to check your work, my first simplification step was not to multiply through both sides by n. Instead, I just distributed terms first, and then did the multiply of both sides by n:

48 - 96/n + 3n - 6 = 70

etc.

 

[buddyboole19]

When you multiply both sides of your original equation by n, you should have gotten:

(n-2) * ((48)+3n) = 70n

i thought when multiplying by the denominator to isolate fractions you had to multiply every term not just the fraction?

why is (n-2) not multiplied by n?

*edit: that worked out - thank you for the help - i thought you had to multiply (n-2) by n as well.. one step and the whole thing is done!

 

[HallsofIvy]

If you multiply ab by n, it is (an)b or a(nb), not (an)(bn)!

 

[Williby]

alright now I am confused, all was good when i was following the original post, then berke wrote his reply and confused me. berk can you show a little more work on how you figured it out please. i got the same results as buddyboole.

 

[Williby]

ok i think i figured out what berk ment by the distribution.
i came up with:
3n^2-28n-96=0 is this right?

i found this math site, but it too is confusing the heck outta me http://www.1728.com/quadr2.htm


encase you haven't noticed its been a while since I've done this kinda math

 

[berkeman]

ok i think i figured out what berk ment by the distribution.
i came up with:
3n^2-28n-96=0 is this right?

i found this math site, but it too is confusing the heck outta me http://www.1728.com/quadr2.htm


encase you haven't noticed its been a while since I've done this kinda math

Yeah, that's correct. And you just use the quadratic formula (the final equation at the bottom of the link you posted) to figure out the answer.