Justification for an Algebraic Expression I have been struggling for the past few days trying to figure out an algebraic expression and solving it.  The question consists of a kid spending ${$}20$ on pens and the options are ${$}4$ , ${$}0.50$ or ${$}0.25$ for a pen. 
This is the equations I have figured out:
$x+y+z=20$
$4a+.50b+.25c=20$
How can I explain how many pens she could buy for ${$}20$ without writing down each and every possible way?  Can I use the formulas? If so, how could I explain that? 

She can buy 0 to 5 of pen a. 
She can buy 0 to (20 - 4i)/.5 = 40 - 8i of pen b if i = the number of pen a she bought.
She then has to buy (40 - 8i)/.25 of pen c if j = the number of pen b she bought.
So I think you are asking how many different ways are there for he to buy pens.
Well there are 6 ways for her to buy pen a and how many pens a she buys determines how many ways there are for her to buy pen b and then there is 1 way for her to buy pen c. 
So the number of ways to buy pens are:
$\sum_{i = 0}^5\sum_{j = 0}^{40 - 8i}1=$
$\sum_{i = 0}^5 40 - 8i+1=$
$41 + 33 + 25 + 17 + 9 + 1 = 126  $ ways to buy pens.
(If she buys 0 a pens then she can buy 0 to 40 pen b so that's 41 ways.
If she buys 1 a pen then she can buy 0 to 32 b pens so that's 33 ways ... and so on.)
