Canteen Fruit Order: Survey Proportions and Discount
A school canteen manager surveys 180 learners about lunch habits. Of the 110 who buy lunch at the canteen, 45 also buy a fruit; of the 70 who bring a packed lunch, 28 buy a fruit. The manager is considering ordering fruit in bulk. Fruit costs M4 each and sells for M7. Unsold fruit is thrown away. From Monday to Thursday, the canteen serves 540 learners in total, and the manager expects the survey proportions to hold. On Friday, a sports event means only 120 learners will be at school, and the manager plans to order fruit for that day using the same overall proportion of fruit buyers. A supplier offers a 10% discount if at least 150 fruits are ordered at once, but delivery takes two days. The manager must decide how many fruits to order for a normal four-day week and whether to use the bulk discount, while keeping expected waste below 15 fruits per day.
1. Use the survey to complete a two-way table with rows 'Buys canteen lunch' and 'Brings packed lunch' and columns 'Buys fruit' and 'Does not buy fruit'. Then find P(buys fruit) and P(buys fruit | buys canteen lunch).
Read the worked answer
Canteen lunch: buys fruit 45, does not 110 − 45 = 65, total 110. Packed lunch: buys fruit 28, does not 70 − 28 = 42, total 70. Column totals: fruit 45 + 28 = 73; no fruit 65 + 42 = 107; grand total 180. P(buys fruit) = 73/180 = 0.406 (3 s.f.). P(buys fruit | buys canteen lunch) = 45/110 = 0.409 (3 s.f.). The two probabilities are close, so buying canteen lunch does not appear to change fruit-buying much in this sample.
2. For the normal four-day week with 540 learners, estimate the number of fruits the manager should expect to sell. If fruit is ordered at full price and exactly that number is ordered, calculate the expected profit.
Read the worked answer
Expected fruit buyers = 540 × (73/180) = 540 × 0.40556 = 219.0, so about 219 fruits. If 219 are ordered and all are sold, cost = 219 × 4 = M876; revenue = 219 × 7 = M1,533; profit = 1,533 − 876 = M657. This assumes demand equals the survey proportion and no waste. If the manager orders more than 219, each extra fruit costs M4 and, if unsold, adds no revenue, so expected profit falls unless demand is genuinely higher.
3. Evaluate the 10% bulk discount for the four-day order and for Friday. The four-day order is 219 fruits; Friday has 120 learners. Should the manager use the discount? Include expected profit and the waste target.
Read the worked answer
Four-day order: 219 fruits qualifies for the discount because 219 ≥ 150. Discounted cost = 219 × 4 × 0.90 = M788.40. If all 219 sell, revenue = M1,533, profit = 1,533 − 788.40 = M744.60, which is M87.60 better than the full-price profit of M657. Friday expected demand = 120 × (73/180) = 48.67, so order about 49 fruits. That is below 150, so no bulk discount applies; cost = 49 × 4 = M196, revenue if all sell = 49 × 7 = M343, profit = M147. To keep waste below 15 fruits per day, ordering close to expected demand is sensible, but the manager should build in a small buffer only if demand variability is known. The bulk discount is worth using for the four-day order, not for Friday alone; combining Friday with another order could reach 150 but risks waste if delivery timing is wrong.