Question 280582
With solution problems like this one, you need to keep track of the 'pure' stuff involved.
The desired result is 60 liters of a 40% alcohol solution, which will have .4(60) = 24 liters of pure alcohol and 60-24 = 36 liters of water.
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The two available solutions to be used in the mixture are a 30% solution and a 70% solution
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x = volume of 30% solution
y = volume of 70% solution
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But we don't need to use two variables because we know that if we have 'x' liters of the 30% solution, then by definition we have 60-x liters of the 70% solution.
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Setting up the equation to solve the problem...
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{{{.3x + .7(60-x) = .4(60) = 24}}}
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multiply through by 10 to get rid of the decimals
{{{3x + 7(60-x) = 4(60)}}}
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collecting and simplifying
{{{3x + 420 - 7x = 240}}}
{{{-4x + 420 = 240}}}
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subtract 420 from both sides
{{{-4x = 240 - 420 = -180}}}
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divide both sides by -4
{{{x = (-180)/(-4) = 180/4 = 45}}}
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So we have 45 liters of 30% solution.
Recall we have 60 liters in total, so we can calculate the volume of 70% solution as:
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{{{x + y = 60}}}
{{{45 + y = 60}}}
{{{y = 15}}}
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Thus our tentative answer is 45 liters of 30% solution + 15 liters of 70% solution will result in 60 liters of 40% solution.
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Of course, we always check our work!
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{{{.3(45) + .7(15) = .4(60) = 24??}}} ??
{{{.3(45) = 13.5}}}
{{{.7(15) = 10.5}}}
{{{13.5 + 10.5 = 24}}}
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Yes!
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Answer:
15 liters of a 70% solution must be mixed with 45 liters of a 30% solution to produce 60 liters of a 40% solution.
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Done