Camp Transport and Drink Packs: Checking Two Claims Before Booking
The P5 cohort is going on a two-day camp. There are 236 pupils and 19 adults (teachers and parent helpers) travelling together, a total of 255 people. The transport company's vans each seat 12 passengers, and the school's assistant principal has already written "21 vans" on the booking form, explaining that 21 × 12 is "about 250, which is close enough". Meanwhile the camp caterer is packing drinks: fruit juice comes in packs of 6 cartons and bottled water comes in packs of 8 bottles. Every camper and adult must receive exactly one juice carton and one bottle of water. The caterer wants to order whole packs only and has suggested 40 packs of juice and 40 packs of water, saying that bulk ordering is cheaper. Before the booking and the order are confirmed, the camp coordinator asks for the numbers to be checked properly.
1. Is 21 vans enough for the trip? Work out the exact number of vans needed and how many seats would be empty, and explain why the assistant principal's reasoning is unsafe.
Read the worked answer
Total travellers = 236 + 19 = 255.
255 ÷ 12 = 21 remainder 3, because 21 × 12 = 252.
So 21 vans carry only 252 people and 3 people are left behind. "Close enough" fails here because people cannot be split or left on the kerb - this is a rounding-up situation, not a rounding-to-nearest situation.
Use 22 vans: 22 × 12 = 264 seats, so 264 − 255 = 9 empty seats.
Note the trap: 255 ÷ 12 = 21.25, which rounds to 21 if you round to the nearest whole number. That is exactly the error on the form.
2. How many packs of juice and how many packs of water should be ordered so that every person gets one of each, and is the caterer's suggestion of 40 packs of each adequate?
Read the worked answer
Juice: 255 ÷ 6 = 42 remainder 3, so 42 packs hold 252 cartons - 3 short. Order 43 packs = 258 cartons, leaving 3 spare.
Water: 255 ÷ 8 = 31 remainder 7, so 31 packs hold 248 bottles - 7 short. Order 32 packs = 256 bottles, leaving 1 spare.
Both answers must be rounded up, because a partial pack cannot be bought and a shortfall means someone goes without.
The caterer's 40 packs of juice give only 240 cartons, which is 15 short of 255. So the bulk suggestion is not merely wasteful - it is insufficient. Water at 40 packs (320 bottles) would be enough but leaves 65 bottles unused.
3. For the staff room the caterer wants equal numbers of juice cartons and water bottles with none left over, using the same pack sizes. What is the smallest number of each drink that allows this, and how many packs of each is that?
Read the worked answer
The number of drinks must be a multiple of both 6 and 8, so find the LCM.
6 = 2 × 3 and 8 = 2³, so LCM = 2³ × 3 = 24.
So 24 cartons and 24 bottles: 24 ÷ 6 = 4 packs of juice and 24 ÷ 8 = 3 packs of water.
Check: 4 × 6 = 24 and 3 × 8 = 24 ✓
Multiples of 24 (24, 48, 72, …) are the only totals that give exact matching with no leftovers, and 24 is the smallest. Any smaller total, such as 18 or 20, cannot be made from both pack sizes at once.