# Generalizing Algebraic Problem

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?

1. How many pens of each kind did she buy?
2. 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.
3. 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?

• What are the possibilities for the $\$4\$ pen? Do all of them work? Do any of them work? Which one(s)? – John Douma Nov 3 '15 at 2:29

## 1 Answer

Hint: You need another equation. Since the number of pens equals 20, you then must also have x+y+z=20. From there, you need to find values of x, y, and z such that both equations are fulfilled.