To determine how many liters of the 5% and 40% solutions the scientist should mix to obtain 10 liters of a 20% acid solution, we can formulate an equation based on the concentration of acid in each solution.
Let's denote:
- \( x \) = liters of the 5% solution
- \( 10 - x \) = liters of the 40% solution (since the total volume needs to be 10 liters)
We know:
- The desired concentration of acid in the final solution is 20%, which means there will be \( 0.20 \times 10 = 2 \) liters of acid in the final solution.
From the two solutions, we can express the total acid in the mixture:
- The 5% solution contributes \( 0.05x \) liters of acid.
- The 40% solution contributes \( 0.40(10 - x) \) liters of acid.
Thus, we can write the equation: \[ 0.20 \times 10 = 0.05x + 0.40(10 - x) \]
This corresponds to option B: \[ (0.20)(10) = 0.05x + 0.40(10 - x) \]
Part B: Solving the Equation
Now let's solve for \( x \):
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Substitute \( 2 \) into the equation: \[ 2 = 0.05x + 0.40(10 - x) \]
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Distribute: \[ 2 = 0.05x + 4 - 0.40x \]
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Combine like terms: \[ 2 = 4 - 0.35x \]
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Rearrange to solve for \( x \): \[ 0.35x = 4 - 2 \] \[ 0.35x = 2 \] \[ x = \frac{2}{0.35} \] \[ x \approx 5.7143 \ (\text{liters of 5% solution}) \]
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Now find the amount of the 40% solution: \[ 10 - x = 10 - 5.7143 \approx 4.2857 \ (\text{liters of 40% solution}) \]
Final Answer (Rounded):
- Liters of 5% solution: \( \approx 5.7 \)
- Liters of 40% solution: \( \approx 4.3 \)
The scientist should mix 5.7 liters of 5% solution with 4.3 liters of 40% solution.
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