Mixing Concentrate at the Juice Counter
A stall at a college fest sells juice by the glass, 250 ml each, at ₹50. Each 20-litre batch of concentrate is blended from two concentrates: A costs ₹180 a litre and B costs ₹120 a litre. The stall's rules are that the blended concentrate must not cost more than ₹140 a litre, and that customers must be able to taste the sharper flavour of A, so A must be at least 25% of every batch. Other costs per batch - cups, ice, helpers - are ₹600. One 20-litre batch fills 80 glasses. Two committee members disagree: one wants as much A as the cost cap allows, the other wants the cheapest blend that satisfies the 25% floor. On the morning of the fest the supplier raises B's price to ₹135 a litre. The stall has already printed its price board and taken advance orders covering 80 glasses per batch.
1. Find the range of A (in litres) per batch that satisfies both rules, and compute the profit per batch at each end of the range.
Read the worked answer
Let x be litres of A, so 20 − x is litres of B.
Cost of blend = 180x + 120(20 − x) = 2,400 + 60x. The cap is ₹140 × 20 = ₹2,800, so 2,400 + 60x ≤ 2,800 gives 60x ≤ 400, i.e. x ≤ 6.67 litres. The taste floor is 25% of 20 = 5 litres.
So 5 ≤ x ≤ 6.67 litres.
Revenue per batch = 80 × 50 = ₹4,000. Other costs ₹600.
- At x = 5: blend cost = 900 + 1,800 = ₹2,700; total ₹3,300; profit ₹700.
- At x = 6.67: blend cost = 1,200 + 1,600 = ₹2,800 (exactly ₹140 a litre); total ₹3,400; profit ₹600.
Every extra litre of A adds ₹60 of cost, so the whole range costs at most ₹100 of profit per batch.
2. Which blend should the stall choose, and how would you justify it to the committee member who wants the cheapest legal blend?
Read the worked answer
Choose the top of the range, about 6.67 litres of A, despite the ₹100 lower profit per batch.
The reason is that the 25% floor is a minimum set to protect taste, not an optimum. At the floor the stall is serving the weakest version of its own product. The ₹100 difference is 2.5% of revenue and 1.7% of the batch's total cost - small against the risk that customers judge the juice thin and do not return during a three-day fest, when repeat buyers are the whole margin.
Framing for the committee: the two rules are a floor and a ceiling, and the ceiling exists because ₹140 is what the stall can afford. Sitting at the ceiling means spending exactly what the pricing allows on the ingredient that drives the product's identity.
A fair qualifier: if sales data showed customers indifferent to A's sharpness, the ₹100 would be free money. The stall has no such data, so the cautious choice is the stronger blend.
3. After B's price rises to ₹135 a litre, can the stall still meet both rules? What should it do?
Read the worked answer
New blend cost = 180x + 135(20 − x) = 2,700 + 45x. The ₹2,800 cap gives 45x ≤ 100, i.e. x ≤ 2.22 litres of A.
But the taste floor needs at least 5 litres of A. Since 2.22 < 5, no blend satisfies both rules - the two constraints now contradict each other.
The cheapest blend that keeps the taste floor is x = 5: cost = 900 + 2,025 = ₹2,925, or ₹146.25 a litre, ₹6.25 above the cap and ₹125 more per batch. Total batch cost = 2,925 + 600 = ₹3,525, so profit at ₹50 a glass falls to ₹475.
Options: raise the glass price to ₹52 (revenue ₹4,160, profit ₹635) for batches after the advance orders are honoured; find a second supplier for B; or accept the ₹146.25 cost and trim other batch costs. What the stall must not do is quietly serve a 25%-A blend at the old cost and pretend the rules still hold.