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Difficulty: HardRatios, Rates, and Proportions

A hydroponics facility uses two nutrient solutions, Solution X and Solution Y. In a standard growth tank, the ratio of the volume of Solution X to the volume of Solution Y is 33 to 55. In a specialized high-yield tank, the ratio of the volume of Solution X to the volume of Solution Y is 44 to 77. A technician mixes 2424 gallons of the mixture from the standard growth tank with 4444 gallons of the mixture from the high-yield tank. What is the ratio of the volume of Solution X to the volume of Solution Y in the final mixture?

  1. A
    7 to 12
  2. 25 to 43Answer
  3. C
    43 to 25
  4. D
    5 to 9

Answer

25 to 43
To find the combined ratio, we must first calculate the actual volume of each solution in both mixtures. For the standard mixture, the ratio of Solution X to Solution Y is 3:53:5, meaning Solution X represents 33+5=38\frac{3}{3+5} = \frac{3}{8} of the total volume, and Solution Y represents 58\frac{5}{8}. In 2424 gallons of this mixture, there are 38×24=9\frac{3}{8} \times 24 = 9 gallons of Solution X and 58×24=15\frac{5}{8} \times 24 = 15 gallons of Solution Y. For the high-yield mixture, the ratio is 4:74:7, so Solution X represents 44+7=411\frac{4}{4+7} = \frac{4}{11} of the total volume, and Solution Y represents 711\frac{7}{11}. In 4444 gallons of this mixture, there are 411×44=16\frac{4}{11} \times 44 = 16 gallons of Solution X and 711×44=28\frac{7}{11} \times 44 = 28 gallons of Solution Y. Combining the mixtures yields a total of 9+16=259 + 16 = 25 gallons of Solution X and 15+28=4315 + 28 = 43 gallons of Solution Y. Therefore, the ratio of the volume of Solution X to the volume of Solution Y in the final mixture is 2525 to 4343.

Step-by-Step Solution

1
Determine the volume of Solution X and Solution Y in the 2424 gallons of standard growth tank mixture.
Solution X volume is 99 gallons; Solution Y volume is 1515 gallons.
The ratio of Solution X to Solution Y is 3:53:5, which means there are 3+5=83 + 5 = 8 total parts. Solution X makes up 38\frac{3}{8} of the mixture, so its volume is 38×24=9\frac{3}{8} \times 24 = 9 gallons. Solution Y makes up 58\frac{5}{8} of the mixture, so its volume is 58×24=15\frac{5}{8} \times 24 = 15 gallons.
2
Determine the volume of Solution X and Solution Y in the 4444 gallons of specialized high-yield tank mixture.
Solution X volume is 1616 gallons; Solution Y volume is 2828 gallons.
The ratio of Solution X to Solution Y is 4:74:7, which means there are 4+7=114 + 7 = 11 total parts. Solution X makes up 411\frac{4}{11} of the mixture, so its volume is 411×44=16\frac{4}{11} \times 44 = 16 gallons. Solution Y makes up 711\frac{7}{11} of the mixture, so its volume is 711×44=28\frac{7}{11} \times 44 = 28 gallons.
3
Calculate the total volumes of Solution X and Solution Y in the final combined mixture.
Total Solution X = 2525 gallons; Total Solution Y = 4343 gallons.
Adding the respective volumes from both mixtures gives: Total Solution X = 9+16=259 + 16 = 25 gallons. Total Solution Y = 15+28=4315 + 28 = 43 gallons.
4
Find the ratio of the total volume of Solution X to the total volume of Solution Y.
The final ratio is 2525 to 4343.
Comparing the total volume of Solution X (2525 gallons) to Solution Y (4343 gallons) yields the ratio 25:4325:43 (or 2525 to 4343).

Key Concept

Combining mixtures with different component ratios by calculating the absolute amounts of each component using part-to-whole fractions.
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Topics

Ratios, Rates, and Proportions

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