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Zorluk: OrtaRatios, Rates, and Percentages

A chemical processing plant prepares a batch of solution by mixing two stock liquid compounds, Solution A and Solution B, to form an initial mixture with a total mass of 480 kg480\text{ kg}. Solution A contains 25%25\% active compound by mass, and Solution B contains 5%5\% active compound by mass. Solution A and Solution B are combined in a mass ratio of 3:13:1, respectively.

To satisfy export specifications, technicians must raise the active compound concentration of the entire mixture to 25%25\% by mass by adding a quantity of pure active compound. How many kilograms of pure active compound must be added to the 480 kg480\text{ kg} mixture?

Cevap: 32 kg

Cevap

32 kg of pure active compound must be added to reach a final concentration of 25% active compound by mass.
The initial mixture consists of 360 kg of Solution A and 120 kg of Solution B. This contains a total of 96 kg of active compound (90 kg from A and 6 kg from B). Adding 32 kg of pure active compound brings the total active compound to 128 kg and total mixture mass to 512 kg, yielding an exact concentration of 128 / 512 = 25%.

Adım Adım Çözüm

1
Calculate the mass of Solution A and Solution B in the initial mixture
Solution A = 360 kg, Solution B = 120 kg
The total mass of 480 kg is divided according to the 3:1 ratio (3/4 and 1/4).
2
Calculate the mass of pure active compound in the initial mixture
Initial active compound = 96 kg
Solution A provides 25% of 360 kg (90 kg) and Solution B provides 5% of 120 kg (6 kg).
3
Set up an equation for the final concentration after adding x kg of active compound
(96 + x) / (480 + x) = 1/4
Adding pure active compound increases both the mass of the active compound and the total mass of the mixture by x kg.
4
Solve for x
x = 32 kg
Cross-multiplying gives 4(96 + x) = 480 + x, which simplifies to 3x = 96.

Anahtar Kavram

Weighted average mixtures and algebraic optimization of concentration ratios
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