Solving a mixture problem with a hidden constraint
A café blends two coffee beans. House blend costs $18/kg, premium costs $32/kg. The owner wants 20 kg of a new blend selling at $24/kg, and sets up 18x + 32(20 − x) = 24 × 20, where x is kilograms of house blend. Solving gives x = 11.43 kg. The owner is pleased until a staff member points out that the premium bean is only available in 5 kg sealed bags and cannot be split, and that the blend must contain at least 6 kg of premium to satisfy a customer's taste requirement. The owner now wants the closest achievable blend to the $24/kg target, using whole bags of premium.
1. Solve the owner's original equation and verify the algebra, then explain why the solution is not usable.
Read the worked answer
18x + 32(20 − x) = 480 18x + 640 − 32x = 480 −14x = −160 x = 160/14 ≈ 11.43
Check: 11.43 kg house at $18 = $205.71; 8.57 kg premium at $32 = $274.29; total $480 for 20 kg = $24/kg. Correct.
It is unusable because premium must come in whole 5 kg bags. 8.57 kg is not a multiple of 5. The continuous model ignored a discrete constraint, so the answer is mathematically right but physically impossible.
2. Find the achievable premium amounts and determine which gives a cost per kilogram closest to $24.
Read the worked answer
Premium must be a multiple of 5 and at least 6, so options within 20 kg are 10, 15, 20 kg (5 kg fails the 6 kg minimum; 0 fails too).
- 10 kg premium, 10 kg house: cost = 10(32) + 10(18) = 320 + 180 = $500, so $25.00/kg.
- 15 kg premium, 5 kg house: cost = 480 + 90 = $570, so $28.50/kg.
- 20 kg premium: $32/kg.
Closest to $24 is 10 kg premium at $25.00/kg, only $1.00/kg above target. The discrete constraint pushes the blend up in cost because the target mix wanted less than 10 kg premium but more than 5 kg.
3. The owner asks whether buying a 1 kg loose scoop of premium (at the same $32/kg) could hit the target exactly. Test this.
Read the worked answer
With loose scoops, premium can be any integer kilogram. We need 18x + 32p = 480 with x + p = 20, giving p = 160/14 ≈ 11.43, not an integer. So no integer-kilogram blend hits exactly $24/kg. Nearest integers: p = 11 gives cost 11(32) + 9(18) = 352 + 162 = $514 → $25.70/kg; p = 12 gives 384 + 144 = $528 → $26.40/kg. So 11 kg premium is closest at $25.70/kg, but it still fails the "at least 6 kg premium" rule? No - 11 ≥ 6, so it satisfies taste. It is closer to target than the 10 kg bag option ($25.00 is actually closer: $1.00 vs $1.70). So the sealed-bag 10 kg option remains best.