Oranges are packed into cartons using box type A (10 oranges each) and box…
2026
Oranges are packed into cartons using box type A (10 oranges each) and box type B (25 oranges each); either type may be used zero or more times. If a carton contains exactly 900 oranges, how many different combinations of the number of type A and type B boxes are possible?
- A.
19
- B.
21
- C.
20
- D.
18
Attempted by 5 students.
Show answer & explanation
Correct answer: A
Concept: The number of non-negative integer solutions of a two-variable linear equation of the form aX + bY = N (where a and b share a common factor) forms an arithmetic progression -- as one variable increases by a fixed step, the other decreases by a fixed step, so the total count of solutions equals the number of terms in that progression.
Let A be the number of type-A boxes (10 oranges each) and B be the number of type-B boxes (25 oranges each).
The packing condition gives the equation 10A + 25B = 900.
Divide throughout by 5, the common factor of 10, 25, and 900: 2A + 5B = 180.
Solve for A: A = (180 - 5B) / 2. For A to be a non-negative integer, (180 - 5B) must be even and at least 0. Since 180 is even, 5B must be even, so B itself must be even.
The largest B can be while keeping A at least 0 comes from 5B <= 180, so B <= 36.
So B takes the even values 0, 2, 4, ..., 36 -- an arithmetic progression with first term 0, common difference 2, and last term 36.
The number of terms in this progression is (36 - 0)/2 + 1 = 19.
Cross-check: at B = 0, A = 90, and 10(90) + 25(0) = 900 -- valid. At B = 36, A = 0, and 10(0) + 25(36) = 900 -- valid. Both boundary cases hold, confirming 19 combinations in total.