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Zorluk: OrtaRatios, Rates, and Proportions

A commercial facility prepares a fruit blend by mixing fruit concentrate with water in a ratio of 3:73:7 by volume. After 15 liters15\text{ liters} of water evaporate from the mixture during processing, the ratio of fruit concentrate to water in the remaining mixture becomes 1:21:2. What was the total volume, in liters, of the original mixture before evaporation?

Cevap: 150 liters

Cevap

The total volume of the original mixture before evaporation was 150 liters.
Represent the initial concentrate volume as 3x3x liters and the initial water volume as 7x7x liters, making the initial total volume 10x10x liters. Evaporating 1515 liters of water leaves 7x−157x - 15 liters of water while the concentrate remains 3x3x liters. Setting the ratio 3x7x−15\frac{3x}{7x - 15} equal to 12\frac{1}{2} yields 6x=7x−156x = 7x - 15, so x=15x = 15. Substituting x=15x = 15 into the total volume expression 10x10x gives 10(15)=15010(15) = 150 liters.

Adım Adım Çözüm

1
Define initial component volumes using ratio multiplier x
Concentrate volume = 3x3x, Water volume = 7x7x, Total volume = 10x10x
The given initial ratio of concentrate to water is 3:73:7.
2
Formulate equation based on water evaporation and the new ratio
3x7x−15=12\frac{3x}{7x - 15} = \frac{1}{2}
Evaporation reduces only the water volume by 15 liters, establishing a new ratio of 1:21:2.
3
Solve the algebraic equation for x
6x=7x−15  ⟹  x=156x = 7x - 15 \implies x = 15
Cross-multiplication simplifies the proportional relationship into a linear equation.
4
Calculate original total volume
10×15=15010 \times 15 = 150 liters
The original total volume is represented by 10x10x.

Anahtar Kavram

Solving component adjustment problems using ratio multipliers
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Konular

Ratios, Rates, and Proportions

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