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A container holds 100 milliliters of a solution that is 25 milliliters acid.
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If 𝑛 milliliters of a solution that is 60 percent acid is added, the function 𝐶 gives the concentration, 50 percent, as a function of the number of milliliters added, where 𝐶 equals 25 plus 0.6𝑛 divided by 100 plus 𝑛.
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Express 𝑛 as a function of 𝐶 and determine the number of milliliters needed to have a solution that is 60 percent acid.
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If we consider the equation 𝐶 equals 25 plus 0.6𝑛 divided by 100 plus 𝑛, then in order to express 𝑛 as a function of 𝐶 we need to rearrange the equation to make 𝑛 the subject.
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Multiplying both sides by 100 plus 𝑛 gives us 𝐶 multiplied by 100 plus 𝑛 equals 25 plus 0.6𝑛.
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Expanding or multiplying out the brackets using the distributive property gives us 100𝐶 plus 𝑛𝐶.
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We now need to collect the like terms.
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Our first step is to subtract a 100𝐶 from both sides of the equation.
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If we then subtract 0.6𝑛 from both sides of the equation, we are left with 𝑛𝐶 minus 0.6𝑛 equals 25 minus 100𝐶.
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At this point, all the terms with 𝑛 in them are on the left-hand side of the equation.
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Factorizing out an 𝑛 gives us 𝑛 multiplied by 𝐶 minus 0.6 equals 25 minus 100𝐶.
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Dividing both sides by 𝐶 minus 0.6 gives us 𝑛 is equal to 25 minus 100𝐶 divided by 𝐶 minus 0.6.
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This is an expression for 𝑛 as a function of 𝐶.
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The second part of our question asked us to determine the number of milliliters needed to have a solution that is 60 percent acid.
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The question also stated that the concentration was equal to 50 percent.
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As 50 percent is equal to 0.5, we can substitute 𝐶 equals 0.5 into our equation.
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This gives us 𝑛 is equal to 25 minus 100 multiplied by 0.5 divided by 0.5 minus 0.6.
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100 multiplied by 0.5 is 50.
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25 minus 50 is equal to negative 25.
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0.5 minus 0.6 is negative 0.1.
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This leaves us with 𝑛 is equal to negative 25 divided by negative 0.1.
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As negative 25 divided by negative 0.1 is equal to 250, the number of milliliters needed is 250 milliliters.
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We need to add 250 milliliters of the solution 𝑛 to ensure we have a solution that is 60 percent acid.