Ratio Problem-Solving - Pre Algebra

1. A recipe for salad dressing calls for oil and vinegar in a 3:1 ratio. Chen makes 2 cups of dressing following this recipe, then adds more oil to make it less acidic. After adding the oil, the ratio becomes 5:1. Chen then decides to bring the ratio back closer to the original by adding some vinegar. If he adds enough vinegar to make the final ratio 4:1, how much total vinegar will be in the final mixture?
   Answer: 0.6 cups of vinegar in the final dressing mixture Original 2 cups with 3:1 ratio: vinegar = $\frac{1}{4} \times 2 = 0.5$ cups, oil = $\frac{3}{4} \times 2 = 1.5$ cups. After adding oil to get 5:1 ratio: vinegar stays 0.5 cups, so oil = $0.5 \times 5 = 2.5$ cups (added $2.5 - 1.5 = 1$ cup oil). After adding vinegar to get 4:1 ratio: oil stays 2.5 cups, so vinegar = $\frac{2.5}{4} = 0.625$ cups. Rounding to one decimal place gives 0.6 cups.
2. A bakery makes muffin batter using flour, sugar, and eggs in a 6:2:1 ratio by weight. They prepare a large batch using 18 pounds of flour. Due to an unexpected large order, they need to make 50% more batter, but they only have 4 more pounds of flour available. If they use all their available flour and maintain the same ratios for the additional batter, how much total sugar will they have used for both batches combined?
   Answer: 7.33 pounds of sugar used in the total baking process First batch with 18 pounds flour (6 parts): each part = $18 \div 6 = 3$ pounds. Sugar for first batch = $2 \times 3 = 6$ pounds. Total first batch = $6 + 2 + 1 = 9$ parts $= 9 \times 3 = 27$ pounds. They want 50% more: $27 \times 1.5 = 40.5$ pounds total, so they need $40.5 - 27 = 13.5$ more pounds of batter. For the additional 13.5 pounds with same ratio (9 parts): each part = $13.5 \div 9 = 1.5$ pounds. This requires $6 \times 1.5 = 9$ pounds flour, but they only have 4 pounds available. With 4 pounds flour (6 parts): each part = $4 \div 6 = \frac{2}{3}$ pounds. Second batch sugar = $2 \times \frac{2}{3} = \frac{4}{3}$ pounds. Total sugar = $6 + \frac{4}{3} = \frac{18 + 4}{3} = \frac{22}{3} \approx 7.33$ pounds.
3. A trail mix recipe combines peanuts, raisins, and chocolate chips in a 5:2:3 ratio by weight. Sarah makes a batch using 2.4 pounds of raisins. She then decides to double the recipe but discovers she only has enough chocolate chips to increase that ingredient by 80%. If she adjusts the peanuts proportionally to match the chocolate chips, how many pounds of raisins will she need for this modified double batch?
   Answer: 4.32 pounds of raisins to maintain the adjusted proportional ratios Original batch: With 2.4 pounds of raisins (2 parts), each part = $2.4 \div 2 = 1.2$ pounds. So peanuts = $5 \times 1.2 = 6$ pounds, chocolate chips = $3 \times 1.2 = 3.6$ pounds. Double recipe would need: peanuts = 12 pounds, raisins = 4.8 pounds, chocolate chips = 7.2 pounds. But chocolate chips can only increase by 80%: $3.6 \times 1.8 = 6.48$ pounds available. If she scales peanuts proportionally to chocolate chips: the chocolate chip scaling factor is $\frac{6.48}{7.2} = 0.9$. So peanuts = $12 \times 0.9 = 10.8$ pounds. The new ratio is approximately $10.8:R:6.48$. To maintain the original 5:2:3 proportions: $\frac{10.8}{5} = \frac{R}{2} = \frac{6.48}{3}$. From chocolate chips: $\frac{6.48}{3} = 2.16$. So $R = 2 \times 2.16 = 4.32$ pounds.

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