Gradient and Cost Model for a Hire Business
A tool-hire business charges a flat call-out fee plus an hourly rate. A customer hiring equipment for 3 hours is charged $155, and for 7 hours is charged $295. The business is considering a new pricing structure: a call-out fee of $50 and an hourly rate of $32, but with a cap so that no hire of 10 hours or more costs more than $340. Separately, a loyalty discount of 12% applies to the total for hires of 5 hours or more. The owner wants to know the break-even hire length at which the new structure becomes cheaper than the old one for short hires, and how the cap and discount interact.
1. Find the call-out fee and hourly rate of the current pricing from the two data points, then write the cost as a linear rule.
Read the worked answer
Let F = call-out fee, r = hourly rate.
3r + F = 155 7r + F = 295
Subtract: 4r = 140, so r = $35 per hour. Then F = 155 − 3×35 = 155 − 105 = $50.
Rule: C = 35h + 50, where h is hours.
Check: h = 7 gives 245 + 50 = $295 ✓. The gradient is the hourly rate ($35/h) and the intercept is the call-out fee ($50). The judgement is recognising that the gradient is the rate of change between the two points, not either charge divided by its hours (155 ÷ 3 = $51.67 would be wrong).
2. Compare the old and new structures for h = 2, 4, 6 and 8 hours, applying the loyalty discount where it applies, and state where the new structure is cheaper.
Read the worked answer
Old: C = 35h + 50. New: C = 32h + 50, with 12% discount for h ≥ 5.
h = 2: Old = 70 + 50 = $120. New = 64 + 50 = $114 (no discount). New cheaper by $6. h = 4: Old = 140 + 50 = $190. New = 128 + 50 = $178 (no discount). New cheaper by $12. h = 6: Old = 210 + 50 = $260. New = 192 + 50 = $242, less 12% = 242 × 0.88 = $212.96. New cheaper by $47.04. h = 8: Old = 280 + 50 = $330. New = 256 + 50 = $306, × 0.88 = $269.28. New cheaper by $60.72.
The new structure is cheaper at every tested length because the call-out fee is identical and the hourly rate is lower. The discount widens the gap once h ≥ 5. The judgement is noticing the equal intercept means the new plan dominates for all h in this range.
3. Apply the cap and discount at h = 11 and h = 13 and explain the order in which the cap and discount should be applied.
Read the worked answer
Raw new cost at h = 11: 32 × 11 + 50 = 352 + 50 = $402. Cap applies (h ≥ 10), so capped at $340. Discount applies (h ≥ 5): 340 × 0.88 = $299.20.
At h = 13: raw = 416 + 50 = $466, capped at $340, discounted to $299.20 - identical to h = 11.
Old structure at h = 11: 385 + 50 = $435; at h = 13: 455 + 50 = $505. The new structure is far cheaper at long hires.
Order matters: applying the discount before the cap gives 402 × 0.88 = $353.76, then capped at $340 - a different and unintended result. The cap should be applied to the undiscounted price first, then the loyalty discount, because the cap is a limit on the list price and the discount is a separate concession. The judgement is that the sequence of operations changes the answer, and the business must define it explicitly.