Question 176305
 Chang bought some 25 cent stamps and some 15 cent stamps and paid $5.00. How many 25 cent stamps did he buy?
<pre><font size = 4  color = "indigo"><b>
Let x = the no. of 25 cent stamps and 
let y = the no. of 15 cent stamps

25x + 15y = 500

5x + 3y = 100

3 is the smallest coefficient in absolute value.
So write 5 and 100 in terms of their nearest multiple
of 3.  So we write 5 as 6-1 and 100 as 99+1.

{{{(6-1)x + 3y = 99 + 1}}}

{{{6x - x + 3y = 99 + 1}}}

Divide every term by 3

{{{2x - x/3 + y = 33 + 1/3}}}

Isolate all the fractions on the right:

{{{2x + y - 33 = x/3 + 1/3}}}

The left side must be an integer, say A,
so the right side must be equal to 
integer A also, so

{{{2x + y - 33 = A}}} and {{{x/3+1/3=A}}}

Clear the second of fractions by multiplying
through by 3:

{{{x+1=3A}}}

Solve for x:

{{{x=3A-1}}}

Substitute in {{{2x + y - 33 = A}}}

{{{2(3A-1)+y-33=A}}}

{{{6A-2+y-33=A}}}

{{{6A+y-35=A}}}

Solve for y

{{{y=35-5A}}}

So now we have

{{{system(x=3A-1,y=35-5A)}}}

Now since he bought some stamps,

{{{x > 0}}} and {{{y>0}}}

So

{{{3A-1>0}}} and {{{35-5A>0}}}

{{{3A>1}}} and {{{-5A>-35}}}

{{{A>1/3}}} and {{{A<7}}}

So since A is an integer {{{1<=A<=6}}}

So there are 6 solutions:

A     x=3A-1   y=35-5A
1       2         30
2       5         25
3       8         20 
4      11         15 
5      14         10
6      17          5 

Edwin</pre>