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Problem of the Week 1

Directions: The problem below is a typical example of an applied problem which we will be solving in this course. Its solution is the standard of quality which I expect for all POWs. Your task is to
  1. neatly copy by hand with no mistakes both the problem statement and its solution below, verbatim, word-for-word onto clean, unused paper. Note: this might be a good time to review the POW presentation criteria in the course syllabus.

  2. visit the AAC on the second floor of Tamarack to get an AAC tutor to sign off on your handwritten copy.

Problem: Billy Bob is mixing up a batch of his famous "Mother Lode Mountain Punch." Here's Billy Bob's secret recipe:

1) Lots of sugar.
2) Several Kool-Aid drink mix packets.
3) Pure mountain spring water.
4) Bourbon (80 proof).

Now, Billy Bob is a bright, well-liked fellow, but he doesn't remember any any algebra, so he needs a little help. Billy Bob is planning a hootenanny and wants to know how much of the mixed Sugar Kool Aid drink (no alcohol) and how much bourbon (40$\%$ alcohol) he needs to mix together to make 5 gallons of Mother Lode Mountain Punch which is 15$\%$ alcohol. Help Billy Bob figure this out; explain how to do it using the techniques you have learned in this course.

Solution: Billy Bob should use the mixture principle to solve this problem. For this particular type of problem, the mixture principle says that the amount of alcohol is the same before and after mixing. In other words, the process of mixing doesn't change the quantity of alcohol present. This is what will enable us form an equation that we can solve.

Let's write an expression for the amount of alcohol before mixing. First, we let $x$ be the unknown amount of Kool-Aid. Then the rest of the 5 gallons which Billy Bob needs is going to be bourbon, that is, we have $5-x$ gallons of bourbon before mixing. Since the Kool-Aid is 0$\%$ alcohol, the amount of pure alcohol contributed by the Kool-Aid is $0\cdot x$, or 0 gallons. And since the bourbon is 40$\%$ alcohol, the amount of pure alcohol contributed by the bourbon is $0.40\cdot (5-x)$ gallons. So the expression for the amount of alcohol before mixing is: $$ \begin{array}{cccc} &\mbox{Alcohol in Kool-Aid}&+&\mbox{Alcohol in bourbon}\\ =&0&+&0.40 \cdot (5-x)\\ =&&&0.40(5-x). \end{array} $$ Now let's talk about the amount of alcohol after mixing. This is actually the easy part since we know that Billy Bob wants 5 gallons of the mixture which he wants to be 15$\%$ alcohol. So after mixing there should be $0.15 \cdot 5=0.75$ gallons of pure alcohol.

We now unleash the mixture principle; the amount of pure alcohol is the same before and after mixing. That is, $$0.40(5-x)=0.75.$$ We now solve this equation: $$ \begin{array}{ccc} 0.40(5-x)&=&0.75\\ &&\\ \frac{0.40}{0.40}(5-x)&=&\frac{0.75}{0.40}\\ &&\\ 5-x&=&\frac{75}{40}\\ &&\\ 5-x&=&\frac{15}{8}\\ &&\\ 5-x&=&1.875\\ &&\\ x-5&=&-1.875\\ &&\\ x&=&5-1.875\\ &&\\ x&=& 3.125.\\ \end{array} $$ Since $x$ is the amount of Kool-aid, Billy Bob needs 3.125 gallons of Kool-Aid for his mixture. The remainder of 5 gallons, that is 1.875 gallons, is bourbon.

Now, we should ask ourselves if these numbers are reasonable. They do seem reasonable. So let's check the numbers themselves to convince ourselves that we really nailed it. Billy Bob wants a mixture which is 15$\%$ alcohol. Since there is no alcohol in the Kool-aid, the bourbon contributes all of the alcohol. Thus, since there is 1.875 gallons of bourbon, and the bourbon is 40$\%$ alcohol, there will be approximately $0.40 \cdot 1.875$, or 0.75 gallons of pure alcohol in the mixture which is $100\cdot \frac{0.75 \mbox{ gallons alcohol }}{5 \mbox{ total gallons of mixture}}= 15\%$. So our answer checks.

Therefore, the answer to our problem of the week is:

Billy Bob needs to mix 3.125 gallons of Kool-Aid and 1.875 gallons of 80 proof bourbon to make a 5-gallon mixture that is 15$\%$ alcohol.

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