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Difficulty: HardRatio and Proportion Word Problems

A pharmaceutical laboratory prepares a liquid vaccine solution by blending three ingredients: Active Compound X, Active Compound Y, and Distilled Water in the volume ratio of 3:4:83 : 4 : 8, respectively. To meet updated formulation guidelines, a chemist adds 1212 liters of Active Compound Y and 3636 liters of Distilled Water to the solution, leaving the quantity of Active Compound X unchanged. If the new ratio of Active Compound X to Distilled Water in the resulting solution is 1:41 : 4, what was the total volume, in liters, of the solution before any ingredients were added?

Answer: 135 liters

Answer

The total volume of the solution before any ingredients were added was 135 liters.
Representing the initial volumes of Active Compound X, Active Compound Y, and Distilled Water as 3x3x, 4x4x, and 8x8x, the initial total volume is 15x15x liters. After adding 3636 liters of Distilled Water while keeping Compound X at 3x3x liters, the ratio of X to Water becomes 3x8x+36=14\frac{3x}{8x + 36} = \frac{1}{4}. Solving 12x=8x+3612x = 8x + 36 gives 4x=364x = 36, so x=9x = 9. Multiplying 1515 by 99 gives the initial total volume of 135135 liters.

Step-by-Step Solution

1
Define the initial component quantities using a common ratio multiplier
The initial volumes of Compound X, Compound Y, and Distilled Water are 3x3x, 4x4x, and 8x8x liters, giving an initial total volume of 15x15x liters.
A ratio of 3:4:83 : 4 : 8 means the actual quantities are integer multiples of a common constant xx.
2
Set up the ratio equation reflecting the additions
The equation comparing the unchanged Compound X to the updated volume of Distilled Water is 3x8x+36=14\frac{3x}{8x + 36} = \frac{1}{4}.
No Compound X was added, so its volume stays 3x3x, whereas 3636 liters were added to the initial 8x8x liters of Distilled Water.
3
Solve the algebraic equation for the multiplier xx
4(3x)=1(8x+36)  ⟹  12x=8x+36  ⟹  4x=36  ⟹  x=94(3x) = 1(8x + 36) \implies 12x = 8x + 36 \implies 4x = 36 \implies x = 9.
Cross-multiplying isolates the terms with xx and allows solving for the multiplier.
4
Calculate the target initial total volume
15×9=13515 \times 9 = 135 liters.
Substituting x=9x = 9 into the initial total volume expression 15x15x yields the final answer.

Key Concept

Solving multi-step ratio word problems by setting up unknown multiplier equations based on partial component alterations
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Topics

Ratio and Proportion Word Problems

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