Field Trip Buses, Budget, and Cost per Person
All 348 fifth graders and 24 teachers are going to a science museum. The district contracts buses that seat 48 passengers each, and no bus may carry more than its capacity. Each bus costs a flat $315. The museum charges $6.25 per student and $9.50 per adult; teachers are counted as adults. The PTA has approved $4,800 for the whole trip, which must cover buses and museum admission. A parent volunteer has already told the principal that "seven buses is plenty" and that "the budget is safe." The principal wants the arithmetic checked before the contract is signed, including a per-person cost to compare with a competing museum's quote of $14.00 per person.
1. How many buses are required? Show why rounding the quotient down would fail.
Read the worked answer
Total passengers: 348 students + 24 teachers = 372 people.
Division: 372 ÷ 48 = 7 remainder 36, because 7 × 48 = 336 and 372 − 336 = 36.
A remainder of 36 is not "almost done" - 36 people still need seats. Seven buses seat only 336, leaving 36 people with no ride. A partly used eighth bus is required.
Answer: 8 buses.
Check: 8 × 48 = 384 seats, and 384 − 372 = 12 empty seats. That is the smallest number of buses that works: 7 is too few, 8 fits, and buying a 9th would add 48 more empty seats for no reason.
The general rule being tested: when people or buses are counted in whole units and something is left over, round up, not to the nearest. Rounding 7.75 to 8 and rounding 7.75 to 7 give different trips - one of them strands 36 children.
2. Compute the total cost of buses plus admission, and judge the volunteer's claim that the budget is safe.
Read the worked answer
Bus cost: 8 × $315 = $2,520 (8 × 300 = 2,400 and 8 × 15 = 120, so 2,400 + 120 = 2,520.)
Student admission: 348 × $6.25 348 × 6 = 2,088 and 348 × 0.25 = 87, so 2,088 + 87 = $2,175
Adult admission: 24 × $9.50 = 24 × 9 + 24 × 0.50 = 216 + 12 = $228
Total: $2,520 + $2,175 + $228 = $4,923
Verdict: the approved amount is $4,800, so the trip is over budget by $4,923 − $4,800 = $123. The volunteer's claim is wrong on both counts - seven buses would strand 36 people, and even the correct eight-bus plan exceeds the approval.
Options the principal could weigh: ask the PTA for $123 more, or cut 20 adults from the trip (20 × $9.50 = $190), which is the only line that can absorb the gap without reducing student seats.
3. Find the cost per person and compare it with the competing museum's $14.00 quote. Show an estimation check as well as the exact value.
Read the worked answer
Exact value: $4,923 ÷ 372 ≈ $13.23 per person.
Work it out with multiplication rather than long division: 372 × 13 = 4,836, which leaves 4,923 − 4,836 = 87. Then 87 ÷ 372 ≈ 0.23, so the quotient is about 13.23. Since 372 × 14 = 5,208 is larger than 4,923, the per-person cost is genuinely below $14.
Estimation check: round 4,923 to 4,800 and 372 to 400 → 4,800 ÷ 400 = 12. That estimate is low because the divisor was rounded up and the dividend was rounded down. A second estimate: 372 × 13 ≈ 370 × 13 = 4,810, close to 4,923, so roughly $13 is right and $14 is a bit too high. Two estimates from different directions bracket $13.23, which is the sign of a sound calculation.
Conclusion: at about $13.23 per person the district's plan is cheaper than the competing quote of $14.00, by roughly $0.77 per person, or about $286 across all 372 travelers. Note the two conclusions are compatible: the plan is over the $4,800 approval and cheaper per person than the alternative. Whether to switch depends on the total the other museum would charge for 372 people, which is 372 × $14.00 = $5,208 - higher than $4,923, so staying with the original museum and finding $123 is the better decision.