Concentrate vs. Pre-Mixed Cleaner: Order Size and Waste
Each school day, custodians across a three-building campus fill 45 spray bottles of general-purpose cleaner, each 32 fluid ounces. The district can buy pre-mixed bottles at $4.25 each, or a 2.5-gallon jug of concentrate at $38.40 and dilute it 1 part concentrate to 64 parts water. Because mixing is done in 5-gallon buckets and crews stop when the bucket is low, only 4.6 gallons of each 5-gallon batch is actually used; the rest is discarded at the end of the shift. The campus operates 180 days per year. Storage is limited to three shelves that each hold six jugs. The supplier offers 10% off the entire concentrate order when at least 12 jugs are purchased. A separate disinfectant is mixed 1:16 and uses about 1 gallon of mixed solution per week for 36 weeks, but that product comes in a different container and is not part of this decision.
1. How many 2.5-gallon jugs of concentrate must the district order for a school year, and what is the annual concentrate cost before any discount? Show the unit conversions and the waste adjustment.
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Convert the daily spray-bottle volume: 45 bottles x 32 fl oz = 1,440 fl oz. Since 128 fl oz = 1 gal, that is 1,440/128 = 11.25 gal of mixed cleaner per day. A 1:64 dilution means 1 part concentrate in 65 parts total mix, so concentrate needed per day is 11.25/65 = 0.1731 gal. The 5-gallon batch waste factor is 5/4.6 = 1.0870, because only 4.6 of every 5 gallons is used. Adjusted concentrate = 0.1731 x 1.0870 = 0.1882 gal/day. For 180 days: 0.1882 x 180 = 33.88 gal. Each jug is 2.5 gal, so 33.88/2.5 = 13.55, which rounds up to 14 jugs. At $38.40 each, cost = 14 x $38.40 = $537.60. The pre-mixed alternative would cost 45 x 180 x $4.25 = $34,425, so concentrate is far cheaper even with waste.
2. Does the 10% bulk discount apply, and does the discounted order fit the storage limit? What is the final cost?
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14 jugs meets the 12-jug threshold, so the 10% discount applies to the whole order. Discounted unit price = $38.40 x 0.90 = $34.56. Total = 14 x $34.56 = $483.84, a saving of $53.76 versus $537.60. Storage capacity is 3 shelves x 6 jugs = 18 jugs, so 14 jugs fit with 4 spaces left. The order is feasible; the discount should be taken.
3. A supervisor proposes switching to 2-gallon buckets, using 1.8 gallons before discarding. Recalculate the number of jugs needed and the discounted cost. Does this change the order decision?
Read the worked answer
With 2-gallon buckets using 1.8 gal, waste factor = 2/1.8 = 1.1111. The no-waste annual concentrate amount is 0.1731 x 180 = 31.15 gal. Adjusted annual concentrate = 31.15 x 1.1111 = 34.62 gal. Jugs = 34.62/2.5 = 13.85, which still rounds up to 14. The discounted cost remains 14 x $34.56 = $483.84. The smaller bucket increases waste slightly but does not change the rounded order or cost, so it is not a reason to switch unless it improves ergonomics or safety. Because the result is near a rounding boundary, the supervisor should recheck exact annual volume before changing the purchasing contract.