Molly went to the store to purchase ink pens. She found three kinds of pens. The first cost $4$ dollars each; the price of the second kind was $4$ for $1$ dollar, and the cost for the third kind was $2$ for $1$ dollar (note it is possible to buy all types of pens individually if desired). She bought $20$ pens and she bought at least one of each kind. The cost was $20$ dollars. When she got back to her office, Molly decided to turn this into a math problem. She asked: Given the cost was $20$ dollars, how many of each ink of pen did I buy?
- How many pens of each kind did she buy?
- Show that the purchase you selected will cost $20$ dollars for $20$ pens and determine whether there is more than one answer to the question or just one answer.
- Provide an explanation that shows how you solved the problem, including the question to whether or not there is more than one answer.
I'm stuck because all I want to do is list out all of the pens. Would an option be $4x+.5y+.25z=20$? Since pens are worth either $\$4.25$ cents or $50$ cents. How can I generalize this?