A shopkeeper offers a scheme "Buy 5 items and get 3 items free". What is the…

Concept

In a "buy m, get n free" scheme the customer pays the marked price of only m items but carries home m + n items. The discount is the marked value the customer never pays for, taken as a fraction of the marked value of everything received: effective discount % = n ÷ (m + n) × 100.

Application

1. Take the marked price of one item as 1 unit, since any common price cancels out of a percentage.

2. Items carried home = 5 paid for + 3 free = 8, so their marked value = 8 units.

3. Amount actually paid = marked price of 5 items = 5 units.

4. Marked value not paid for = 8 - 5 = 3 units.

5. Effective discount = (3 ÷ 8) × 100 = 37.5%.

Cross-check

Substituting m = 5 and n = 3 straight into the formula gives 3 ÷ (5 + 3) × 100 = 37.5%, reproducing the step-by-step result. The other percentages on offer correspond to different bases:

• 3 ÷ 5 × 100 = 60% measures the free items against the items paid for, so its base is the money spent rather than the marked value of the goods received.

• 3 ÷ 10 × 100 = 30% needs a ten-item total, which is the "buy 7, get 3 free" scheme.

• 3 ÷ 9 × 100 = 33.33% needs a nine-item total, which is the "buy 6, get 3 free" scheme.

Result: the effective discount is 37.5%.