The Break-Even Decision at the Print Workshop
A small print workshop is choosing between two pricing plans for a bulk job of booklets. Plan A charges a fixed setup fee of 1,200 taka plus 18 taka per booklet. Plan B charges no setup fee but 30 taka per booklet, with a discount clause: once the order exceeds 200 booklets, the per-booklet rate on the entire order drops to 26 taka. The client's order size n is not yet fixed but is known to be a whole number between 100 and 400 inclusive. The workshop owner wants to know, for each possible range of n, which plan is cheaper, and she is worried that the discount clause might create a sudden jump in cost that she has not accounted for. She also wants a single algebraic rule she can apply without checking cases by hand every time.
1. Express the total cost of each plan as a function of n, and find all order sizes in 100 ≤ n ≤ 400 for which Plan A is strictly cheaper than Plan B.
Read the worked answer
Write the costs:
- A(n) = 1200 + 18n (for all n)
- B(n) = 30n for 100 ≤ n ≤ 200; B(n) = 26n for 200 < n ≤ 400.
Case 1: 100 ≤ n ≤ 200. Solve 1200 + 18n < 30n → 1200 < 12n → n > 100. So for 101 ≤ n ≤ 200, Plan A is strictly cheaper; at n = 100 they tie (1200 + 1800 = 3000 = 30·100).
Case 2: 200 < n ≤ 400. Solve 1200 + 18n < 26n → 1200 < 8n → n > 150. This holds for every n in this case.
So Plan A is strictly cheaper for all whole n with 101 ≤ n ≤ 400, and the plans tie only at n = 100.
Note the boundary: at n = 200, B = 6000 and A = 4800, so A wins; at n = 201, B = 26·201 = 5226 and A = 4818, so A still wins - the discount does not overturn the conclusion.
2. Examine whether Plan B's cost function is continuous at n = 200, and explain what this means for the owner's worry about a sudden jump.
Read the worked answer
Evaluate B just below and just above 200:
- Left limit: as n → 200⁻, B = 30n → 6000.
- Right limit: as n → 200⁺, B = 26n → 5200.
- B(200) = 30·200 = 6000.
The one-sided limits differ (6000 vs 5200), so B is discontinuous at n = 200 - a downward jump of 800 taka. In practical terms, ordering 201 booklets costs 5226, less than ordering 200 at 6000. The jump is a fall, not a rise, so it never makes Plan B unexpectedly expensive; if anything it makes B more attractive at the boundary, yet the calculation above shows A still wins there.
The owner's algebraic rule: compare 1200 + 18n with the applicable B rate. Since A's margin grows with n in both regimes, she only needs to check the tie point n = 100 and the regime boundary n = 200.