Comparing Two Supermarket Unit Prices with a Shrinking Pack
A shopper compares two packages of the same laundry detergent. Brand A sells a 1.5 kg box for $7.80 but has recently shrunk from 1.8 kg at the same price. Brand B sells a 2.4 kg box for $11.04. A store loyalty card gives 15% off the marked price of Brand A only, applied at the till. The shopper does laundry about 5 times a week and uses an estimated 60 g per load. She wants to know the cost per kilogram under each option, the cost per load, and how many weeks a single box of the cheaper-per-load brand would last. She also wants to know whether the shrink from 1.8 kg to 1.5 kg changed the effective price increase as a percentage.
1. Compute the cost per kilogram for Brand A with the discount and for Brand B without it, and identify the better unit price.
Read the worked answer
Brand A marked price $7.80 for 1.5 kg. With 15% off, the shopper pays 85% of $7.80 = 0.85 x 7.80 = $6.63.
Cost per kg = 6.63 / 1.5 = $4.42 per kg.
Brand B: 11.04 / 2.4 = $4.60 per kg.
Check B by scaling: 2.4 kg at $4.60/kg gives 11.04, consistent.
Brand A is cheaper per kilogram by $0.18/kg, about a 3.9% advantage (0.18/4.60). Note that without the loyalty discount Brand A would be 7.80/1.5 = $5.20/kg, which is worse than Brand B. The discount, not the pack, drives the decision.
2. Estimate the cost per load for each brand and the number of weeks one Brand A box lasts at 5 loads per week.
Read the worked answer
Cost per load = cost per kg x kg per load. At 60 g = 0.06 kg per load:
Brand A: 4.42 x 0.06 = $0.2652, about 27 cents per load. Brand B: 4.60 x 0.06 = $0.2760, about 28 cents per load.
Loads per Brand A box = 1.5 kg / 0.06 kg = 25 loads.
At 5 loads per week, 25/5 = 5 weeks per box.
Brand B would give 2.4/0.06 = 40 loads, or 8 weeks.
The per-load gap is under a cent, so the choice depends on whether the shopper values the lower total outlay ($6.63 vs $11.04) or fewer shopping trips. The judgement is that unit price alone does not settle it once pack size and trip frequency matter.
3. The 1.8 kg box was previously sold at $7.80. Express the effective price increase caused by the shrink to 1.5 kg as a percentage, and comment on how this compares with a headline 'same price' claim.
Read the worked answer
Old unit price = 7.80/1.8 = $4.333/kg. New unit price = 7.80/1.5 = $5.20/kg.
Percentage increase = (5.20 - 4.333)/4.333 x 100 = 0.8667/4.333 x 100 = 20%.
Shortcut: the quantity fell from 1.8 to 1.5, a factor of 1.5/1.8 = 0.8333, i.e. a 16.7% reduction in product. Price per kg rises by 1/0.8333 - 1 = 0.20, or 20%.
The headline 'same price' hides a 20% unit-cost increase, which is larger than the 16.7% quantity cut because the percentage is taken against the smaller base. The shopper should compare unit prices, not shelf prices, and should note that even with the 15% loyalty discount the new pack ($4.42/kg) is only just below the old undiscounted unit price ($4.33/kg is actually lower).