Hiring chairs: comparing two delivery-plus-per-chair charges
Ms. Bacchus is on the committee organising a school's 50th-anniversary dinner. Two companies rent chairs. Comfort Seating charges a delivery fee of $1,200 plus $80 for each chair. Quick Chairs charges $300 delivery plus $110 for each chair. Both deliver any number of chairs, and there is no other charge. The committee expects between 25 and 45 guests and wants one chair per guest plus 3 spare chairs for the head table. Guests have not yet been confirmed, so the committee is testing several numbers before deciding. The treasurer has $5,000 for chairs and does not want to be embarrassed by running out on the night. A committee member says, 'Quick Chairs has the smaller delivery fee, so it must be cheaper for a small dinner.'
1. Write a rule for the total cost of n chairs from each company, and find the number of chairs for which both companies charge the same amount.
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Rules
- Comfort: C = 1,200 + 80n
- Quick: Q = 300 + 110n
Here n is the number of chairs; the number in front of n is the cost per chair and the other number is the fixed delivery charge that does not change with n.
Setting them equal 1,200 + 80n = 300 + 110n Subtract 80n from both sides: 1,200 = 300 + 30n Subtract 300: 900 = 30n n = 30
Check: Comfort = 1,200 + 80(30) = 1,200 + 2,400 = $3,600. Quick = 300 + 110(30) = 300 + 3,300 = $3,600. Both agree.
At 30 chairs the two companies cost the same, $3,600. Below 30, Quick's low delivery fee wins because Comfort's large fixed charge has not yet been spread over enough chairs; above 30, Comfort's cheaper per-chair rate overtakes it.
2. The committee settles on 40 guests. How many chairs are needed, which company is cheaper, and how many chairs could they afford from that company with $5,000?
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Chairs needed: 40 guests + 3 spares = 43 chairs.
Costs for 43 chairs
- Comfort: 1,200 + 80 × 43 = 1,200 + 3,440 = $4,640
- Quick: 300 + 110 × 43 = 300 + 4,730 = $5,030
Comfort is cheaper by $390, and Quick is over the $5,000 budget. Note that 43 is above the crossover of 30, exactly as the rules predict.
Maximum chairs within $5,000 (Comfort) 1,200 + 80n ≤ 5,000, so 80n ≤ 3,800, giving n ≤ 47.5. Chairs must be a whole number and rounding up would break the budget, so n = 47. Check: 1,200 + 80 × 47 = 1,200 + 3,760 = $4,960, leaving $40. Booking 48 would cost $5,040, which is $40 too much.
3. Respond to the committee member's claim that 'the smaller delivery fee means Quick Chairs is cheaper for a small dinner'. Use two or three chair numbers to support your answer.
Read the worked answer
The claim is only half right. A small delivery fee helps, but it is not the whole cost.
- 25 chairs: Comfort = 1,200 + 2,000 = $3,200; Quick = 300 + 2,750 = $3,050. Quick is cheaper by $150.
- 30 chairs: both $3,600 - a tie.
- 35 chairs: Comfort = 1,200 + 2,800 = $4,000; Quick = 300 + 3,850 = $4,150. Comfort is cheaper by $150.
The reason is the trade-off between the two charges. Quick's $110 per chair is $30 more than Comfort's $80, so every chair closes $30 of the $900 gap in the delivery fees. After 30 chairs (900 ÷ 30) the gap is gone and Comfort is ahead. The member has compared only one of the two charges; the correct comparison is total cost at the actual number of chairs, and the committee's 43 chairs falls on Comfort's side of the crossover.