Question Video: Writing and Evaluating Rational Functions in a Real-World Context

A container holds 100 mL of a solution that is 25 mL acid. If 𝑛 mL of a solution that is 60% acid is added, the function 𝐢 gives the concentration, 50%, as a function of the number of milliliters added, 𝐢 = (25 + 0.6𝑛)/(100 + 𝑛). Express 𝑛 as a function of 𝐢 and determine the number of milliliters needed to have a solution that is 60% acid.

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Video Transcript

A container holds 100 milliliters of a solution that is 25 milliliters acid. If 𝑛 milliliters of a solution that is 60 percent acid is added, the function 𝐢 gives the concentration, 50 percent, as a function of the number of milliliters added, where 𝐢 equals 25 plus 0.6𝑛 divided by 100 plus 𝑛. Express 𝑛 as a function of 𝐢 and determine the number of milliliters needed to have a solution that is 60 percent acid.

If we consider the equation 𝐢 equals 25 plus 0.6𝑛 divided by 100 plus 𝑛, then in order to express 𝑛 as a function of 𝐢 we need to rearrange the equation to make 𝑛 the subject. Multiplying both sides by 100 plus 𝑛 gives us 𝐢 multiplied by 100 plus 𝑛 equals 25 plus 0.6𝑛. Expanding or multiplying out the brackets using the distributive property gives us 100𝐢 plus 𝑛𝐢.

We now need to collect the like terms. Our first step is to subtract a 100𝐢 from both sides of the equation. If we then subtract 0.6𝑛 from both sides of the equation, we are left with 𝑛𝐢 minus 0.6𝑛 equals 25 minus 100𝐢.

At this point, all the terms with 𝑛 in them are on the left-hand side of the equation. Factorizing out an 𝑛 gives us 𝑛 multiplied by 𝐢 minus 0.6 equals 25 minus 100𝐢. Dividing both sides by 𝐢 minus 0.6 gives us 𝑛 is equal to 25 minus 100𝐢 divided by 𝐢 minus 0.6. This is an expression for 𝑛 as a function of 𝐢.

The second part of our question asked us to determine the number of milliliters needed to have a solution that is 60 percent acid. The question also stated that the concentration was equal to 50 percent. As 50 percent is equal to 0.5, we can substitute 𝐢 equals 0.5 into our equation. This gives us 𝑛 is equal to 25 minus 100 multiplied by 0.5 divided by 0.5 minus 0.6.

100 multiplied by 0.5 is 50. 25 minus 50 is equal to negative 25. 0.5 minus 0.6 is negative 0.1. This leaves us with 𝑛 is equal to negative 25 divided by negative 0.1. As negative 25 divided by negative 0.1 is equal to 250, the number of milliliters needed is 250 milliliters. We need to add 250 milliliters of the solution 𝑛 to ensure we have a solution that is 60 percent acid.

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