Stationery Drive: Two Box Sizes, One Total, One Budget
The P5 classes are running a stationery drive. Pencils arrive from the supplier in two box sizes only: large boxes holding 24 pencils and small boxes holding 10 pencils. At the end of the drive, the volunteers count 30 boxes in total and 552 pencils. The class treasurer, who has not opened any boxes, claims there must be at least 20 large boxes "because 552 is a big number and large boxes hold the most". A large box costs the school $7.20 and a small box costs $3.50; the drive's budget was $180. Before the thank-you letters are written, the organiser wants the exact number of each box size, and also wants to know whether the treasurer's claim holds and whether the order stayed inside the budget.
1. Find the exact number of large boxes and small boxes, showing your reasoning and a check.
Read the worked answer
Supposition method. Suppose all 30 boxes were small: 30 × 10 = 300 pencils.
Actual total is 552, so 552 − 300 = 252 pencils are unaccounted for.
Each time one small box is swapped for one large box, the pencil count rises by 24 − 10 = 14.
252 ÷ 14 = 18 swaps, so there are 18 large boxes and 30 − 18 = 12 small boxes.
Check: 18 × 24 = 432; 12 × 10 = 120; 432 + 120 = 552 ✓, and 18 + 12 = 30 ✓
Algebra gives the same result: 24L + 10(30 − L) = 552 → 14L = 252 → L = 18.
2. Is the treasurer's claim of "at least 20 large boxes" correct? Justify your answer with a calculation.
Read the worked answer
Test 20 large boxes. That leaves 30 − 20 = 10 small boxes.
20 × 24 + 10 × 10 = 480 + 100 = 580 pencils, not 552.
580 is too many, and any count above 20 large boxes gives even more pencils, so 20 or more large boxes is impossible. The true count is 18, so the claim is false.
The treasurer's reasoning fails because "a big total" does not force most boxes to be large - the small boxes still contribute. Because each large box adds only 14 pencils over a small one, the answer has to be solved from both conditions at once, not guessed from the size of the total.
3. Did the drive stay within its $180 budget, and which box size is better value per pencil?
Read the worked answer
Cost: 18 × $7.20 = $129.60; 12 × $3.50 = $42.00.
Total = 129.60 + 42.00 = $171.60, which is under $180, leaving $8.40 unspent.
Value per pencil:
- Large: $7.20 ÷ 24 = $0.30 per pencil
- Small: $3.50 ÷ 10 = $0.35 per pencil
The large box is 5 cents cheaper per pencil, so it is better value - worth knowing if the drive runs again, although the mix is constrained by how many of each box the supplier provides. Overall cost per pencil is 171.60 ÷ 552 ≈ $0.31.