5n + 10d + 25q = 100
Divide through by 5
n + 2d + 5q = 20
n = 20 - 2d - 5q
0 < n < 20
0 < 20 - 2d - 5q < 20
-20 < -2d - 5q < 0
20 > 2d + 5q > 0
That has the graph of all the lattice points (d,q) (where d and
q are non-negative integers on or inside the triangle whose
corners (d,q) are (0,0), (10,0) and (0,4)
The graph below has x as d and y as q.
Now we find how many lattice points (d,q) are on the line
2d + 5q = 20
Write 5q as 4q + q
2d + 4q + q = 20
Divide through by 2
d + 2q + q/2 = 10
q/2 = 10 - d - 2q
The right side is an integer, and so is the left side.
The left side is non-negative so both sides are
non-negative integers, say A,
q/2 = A
q = 2A
10 - d - 2q = A
10 - d - 2(2A) = A
10 - d - 4A = A
10 - 5A = d
0 < d < 10
0 < 10-5A < 10
0 < 2 - A < 2
-2 < -A < 0
2 > A > 0
Thus A = 0,1,2
That means that there are 3 lattice points (d,q) on the line
d = 10-5A, q = 2A
(10,0), (5,2), (0,4)
The number of lattice points in or on the rectangle with corners
(0,0), (10,0), (10,4), (0,4)
is 11x5 = 55 (that is, there are 11 integers from 0 through 10 inclusive,
and 5 integers from 0 through 4 inclusive.
There are 3 lattice points on the diagonal. So there are 55-3= 52
lattice points in or on the rectangle that are not on the diagonal,
and since there are as many lattice points above the diagonal as
there are below the line, there are 26 latice points below the line.
Adding the 3 lattice points on the diagonal gives us 29 lattice points.
Thus there are 29 solutions.
1. 0 quarters, 0 dimes, and 20 nickels.
2. 0 quarters, 1 dimes, and 18 nickels.
3. 0 quarters, 2 dimes, and 16 nickels.
4. 0 quarters, 3 dimes, and 14 nickels.
5. 0 quarters, 4 dimes, and 12 nickels.
6. 0 quarters, 5 dimes, and 10 nickels.
7. 0 quarters, 6 dimes, and 8 nickels.
8. 0 quarters, 7 dimes, and 6 nickels.
9. 0 quarters, 8 dimes, and 4 nickels.
10. 0 quarters, 9 dimes, and 2 nickels.
11. 0 quarters, 10 dimes, and 0 nickels.
12. 1 quarters, 0 dimes, and 15 nickels.
13. 1 quarters, 1 dimes, and 13 nickels.
14. 1 quarters, 2 dimes, and 11 nickels.
15. 1 quarters, 3 dimes, and 9 nickels.
16. 1 quarters, 4 dimes, and 7 nickels.
17. 1 quarters, 5 dimes, and 5 nickels.
18. 1 quarters, 6 dimes, and 3 nickels.
19. 1 quarters, 7 dimes, and 1 nickels.
20. 2 quarters, 0 dimes, and 10 nickels.
21. 2 quarters, 1 dimes, and 8 nickels.
22. 2 quarters, 2 dimes, and 6 nickels.
23. 2 quarters, 3 dimes, and 4 nickels.
24. 2 quarters, 4 dimes, and 2 nickels.
25. 2 quarters, 5 dimes, and 0 nickels.
26. 3 quarters, 0 dimes, and 5 nickels.
27. 3 quarters, 1 dimes, and 3 nickels.
28. 3 quarters, 2 dimes, and 1 nickels.
29. 4 quarters, 0 dimes, and 0 nickels.
Edwin
