To solve the problem, we need to find out how many liters of the 32% acid solution to add to the 50 liters of the 22% acid solution in order to create a final solution that is 28% acid.
Let \( x \) be the amount of 32% acid solution to be added in liters.
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Calculate the amount of acid in the 22% solution: \[ \text{Acid from 22% solution} = 0.22 \times 50 = 11 \text{ liters} \]
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Calculate the amount of acid in the 32% solution: \[ \text{Acid from 32% solution} = 0.32 \times x \text{ liters} \]
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Total acid in the final solution: The total amount of acid after adding \( x \) liters of the 32% solution will be: \[ \text{Total acid} = 11 + 0.32x \text{ liters} \]
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Total volume of the final solution: The total volume of the mixture will be: \[ \text{Total volume} = 50 + x \text{ liters} \]
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Set up the equation for desired concentration: We want the final concentration to be 28%. Therefore, we set up the equation: \[ \frac{11 + 0.32x}{50 + x} = 0.28 \]
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Multiply both sides by \( 50 + x \) to eliminate the fraction: \[ 11 + 0.32x = 0.28(50 + x) \]
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Expand the right side: \[ 11 + 0.32x = 14 + 0.28x \]
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Combine like terms: Subtract \( 0.28x \) from both sides: \[ 11 + 0.32x - 0.28x = 14 \] Simplifying gives: \[ 11 + 0.04x = 14 \]
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Isolate \( x \): \[ 0.04x = 14 - 11 \] \[ 0.04x = 3 \] \[ x = \frac{3}{0.04} = 75 \]
Thus, the teaching assistant should add 75 liters of the 32% acid solution to the 22% acid solution to obtain a 28% acid solution.