In how many ways is it possible to spend exactly $200 on pumpkins if large pumpkins cost $5 and small $3

Answer 1

well, 40 $5 pumpkins cost exactly $200
LCM(5,3) = 15
So, for every 15/5=3 fewer $5/pumpkins, you get 15/3=5 more $3 pumpkins.
So, we have
$5 $3
----------
40 0
37 5
34 10
31 15
28 20
25 25
22 30
19 35
16 40
13 45
10 50
7 55
4 60
1 65