Mr. Hill had 8 bottles of apple juice and 48 bottles of orange juice in the end. This bar model shows that if we start with 3:2 and want 1:6 we can rewrite 3:2 as 9:6 so the 6’s match. This means the 8 boxes on the left that total 64 each represent 8 bottles so that the original 3:2 was really 72:48. After he sells 64 of the 72, he has 8 left and still has 48 orange juice bottles which simplifies to 1:6.

Q: What are all the possible solutions to sum of consecutive positive integers that sum to 2025?

A: There are many solutions if we take out the word “longest”.
1 number 2025
2 numbers 1012 + 1013
3 numbers 674 + 675 + 676
5 numbers 403 + 404 + 405 + 406 + 407
6 numbers 335 + 336…+ 340

9 numbers 221 + 222 + …+ 299
10 numbers 198 + 199 + 200 + … +207
15 numbers 128 + 129 + … + 142
18 numbers 104 + 105 + … + 121
25 numbers 69 + 70 + …+ 93

27 numbers 62 + 63 + … + 98
30 numbers 53 + 54 + … + 81+ 82
45 numbers 22 + 23 + …67+ 68
54 numbers 11 + 12 + ….+ 63 + 64