Discount and Credit: Does the Order of Operations Change the Total?
A bakery sells celebration cakes at list price p dollars. For corporate orders, the bakery applies a 20% discount, then subtracts a flat $5 loyalty credit from the discounted price, and only then adds 8% sales tax on the resulting amount. A rival caterer quotes the same list prices but subtracts the $5 credit first, applies the 20% discount to that reduced price, and then adds the same 8% tax. The bakery's owner suspects the two sequences give the same answer, since both involve one percent change and one flat reduction, and wants to know before printing a comparison sheet. The bakery never lets a total fall below zero, so the credit applies in full only when the discounted price is at least $5.
1. Write a function for each company's total as a function of list price p, and evaluate both at p = $25.
Read the worked answer
Bakery (discount, then credit, then tax): B(p) = 1.08(0.80p − 5)
Caterer (credit, then discount, then tax): C(p) = 1.08 · 0.80(p − 5) = 1.08(0.80p − 4)
At p = 25:
- B(25) = 1.08(20 − 5) = 1.08 × 15 = $16.20
- C(25) = 1.08(0.80 × 25 − 4) = 1.08(20 − 4) = 1.08 × 16 = $17.28
The two sequences are not equivalent. The discount multiplies the credit in the caterer's version (0.80 × 5 = 4), so the customer loses $1 of the credit before tax.
2. Which company charges less, and by how much? Show whether the gap depends on the list price.
Read the worked answer
Subtract: C(p) − B(p) = 1.08(0.80p − 4) − 1.08(0.80p − 5) = 1.08[(0.80p − 4) − (0.80p − 5)] = 1.08 × 1 = $1.08
The p terms cancel, so the gap is a constant $1.08 - the bakery is cheaper for every price where the credit applies. The 8% tax inflates the $1 difference in credit to $1.08.
Check at p = 25: 17.28 − 16.20 = $1.08 ✓
Why the cancellation is expected: taking 20% off and then subtracting a constant does not commute with subtracting the constant and then taking 20% off. Applying the percent first preserves the full $5 credit; applying it second shrinks the credit to $4.
3. A corporate customer paid exactly $16.20 at the bakery. What was the list price? Also state the price range over which the owner's comparison sheet is valid.
Read the worked answer
Undo the steps in reverse order.
- Remove tax: 16.20 ÷ 1.08 = $15.00
- Add back the credit: 15.00 + 5 = $20.00
- Undo the discount: 20.00 ÷ 0.80 = $25.00
Validity range: the flat credit assumes the discounted price is at least $5, i.e. 0.80p ≥ 5, so p ≥ $6.25. Above that threshold the constant $1.08 gap holds. Below $6.25 the bakery's total would go negative and the no-negative rule kicks in, so the function stops describing reality - the comparison sheet should be labeled for list prices of $6.25 and up.
For the caterer the analogous floor is p ≥ $5, since the credit is taken before the discount. The two floors differ, which is another reason a single blanket statement is unsafe.