Trail-Mix Bag Limit and Almond Purchase
The seventh-grade field trip committee is packing 180 individual trail-mix bags, each holding 250 grams. The mix must use raisins, almonds, and pretzels in a 3:2:5 ratio by weight. The pantry has 18 kg of raisins, 12 kg of almonds, and 20 kg of pretzels. The committee wants to make as many full 250-gram bags as possible without buying more ingredients, and every bag must have the same ratio. A parent asks what percent of each bag is almonds. For the almonds, two suppliers offer these prices: Supplier A sells 5 kg for $42, and Supplier B sells 8 kg for $60. The committee will buy only the almonds they still need and wants the lower total cost, even if that means buying a larger package than necessary. They must report the limiting ingredient and the number of bags they can actually fill.
1. Determine the maximum number of full bags. Show the required amount of each ingredient for 180 bags and identify the limiting ingredient.
Read the worked answer
For 180 bags, the total mix is 180 × 0.25 kg = 45 kg. The ratio 3:2:5 has 3 + 2 + 5 = 10 parts, so each part is 45 ÷ 10 = 4.5 kg. Raisins needed: 3 × 4.5 = 13.5 kg; almonds: 2 × 4.5 = 9 kg; pretzels: 5 × 4.5 = 22.5 kg. The pantry has 18 kg raisins, 12 kg almonds, and 20 kg pretzels. Pretzels are limiting because 20 kg is less than 22.5 kg. With 20 kg pretzels for 5 parts, each part is 4 kg, so total mix is 10 × 4 = 40 kg. That makes 40 ÷ 0.25 = 160 full bags. The needed raisins are 12 kg and almonds 8 kg, both available, so 160 bags is possible.
2. What percent of a correctly made bag is almonds? Explain why the answer is not 2/5.
Read the worked answer
The whole mix has 10 equal parts, and almonds are 2 of those parts. The almond fraction is 2/10 = 0.20 = 20%. The fraction 2/5 would compare almonds only to raisins plus almonds, or would treat the total as 5 parts. The correct denominator is the total number of ratio parts, 3 + 2 + 5 = 10, so 20% is the percent of the bag that is almonds.
3. The committee needs 8 kg of almonds for 160 bags. Which supplier gives the lower total cost? Show unit prices and total costs.
Read the worked answer
Supplier A: $42 ÷ 5 kg = $8.40 per kg. Buying 8 kg requires two 5-kg packages, or 10 kg, for 2 × $42 = $84. Supplier B: $60 ÷ 8 kg = $7.50 per kg. One 8-kg package costs exactly $60 for the needed 8 kg. Supplier B has the lower unit price and the lower total cost: $60 versus $84. The committee should buy from Supplier B, assuming the extra 2 kg from Supplier A is not useful.