Math 111-04PQ 2.6Feb. 28, 2014Name:You are a restaurant bartender and your boss has asked you to take a certain mixed, al-coholic drink served at the restaurant and increase its alcohol content, to meet customerrequests, by adding an amount of a clear, relatively flavorless 90% alcohol liquour car-ried by the restaurant. The mixed drink is currently 20% alcohol, and your boss wouldlike to increase it to 30% by adding some of the 90% alcohol product. How many ouncesof the 90% solution should you add to one ounce of the original 20% alcohol mixed drinkin order to put the alcohol concentration of the resulting mixture at 30%?Answer and pep-talk: One legitimately frustrating thing about word problems ingeneral was also one which I think bothered almost everyone in the class: trying to figureout exactly what is being asked about and then, after that, wondering if one knows howsuch-and-such a physical scenario is “supposed” to be written down mathematically.This, anyway, is how I would articulate the difficulty which I also have experiencedwith word problems. There is lots to think about and consider around this issue!Anyway, here is a basic breakdown of how to go about solving this problem. Maybethe first thing, though, is to arrive at the conviction that there is a solution and so theremust be some way to find it.(a) First, before proceeding any further—to prevent yourself from proceeding under astate of doubt and anxiety—settle on a definition of the terms being asked about.You think to yourself, “What does ‘concentration’ mean; what is its definition?”And, then, second, you also worry, “Now, is my definition going to be the one whichI am supposed to use; is it the ‘official’ one?” But, really, I am guessing that yourown personal way of explaining what concentration means is the same thing whichI mean by it and what others mean by it: the amount of some ingredient given as aproportion of the whole. Maybe you wouldn’t use that exact wording, but I thinkyou would probably have the same underlying concept in mind? Another way toput it is “Concentration of a particular ingredient in some sort of mixture is theamount of that ingredient (in whatever units) divided by the total amount of allingredients (in whatever units) in the mixture.”(b) The next step is to figure out how you write down your thought mathematically.First, let me rephrase it in a different way, which seems a more straightforward wayof explaining “concentration.” How about: “the amount of ingredient (measured inwhatever units) present in one unit of the other stuff into which it is mixed.” Perhapsthe ingredient is mixed in uniformly, or perhaps the concentration is interpreted asan average—either way is fine, and in either case, at some point a lightbulb maypop on for you: the way to calculuate it is to take the amount of ingredient inwhatever amount of the mixture and divide by that latter amount. (One way tosay it: Concentration is a rate, i.e.: a ratio.) So, let C stand for concentration ofthe ingredient, A stand for the amout of the ingredient, and T stand for the totalamount of stuff T . ThenC =AT.It helps to think of some examples. The concentration of alcohol in some portionof alcoholic drink is given by doing:conc. of alcohol =vol. of alcohol in mixturetotal vol. of mixture.The percentage of alcohol, of course, is got from that by multiplying by 100. Theproof, on the other hand, is got by multiplying the concentration by 200 (for what-ever reason).Another case of concentration, which is a little bit different, is one in which theunits of the numerator and denominator do not agree. For instance, you can givethe concentration of salt in a solution of salt and water in units of grams per liter;if I say I have some salt water which has a concentration of 16 grams per liter, itmeans, of course, that for any liter I draw out of the bucket (or whatever) therewill be 16 grams of salt diffused in it in some way.(c) Now, to solve the problem, you go on the hunch that, indeed, there is a way to getthat 30% concentration of alcohol by adding in the stronger liquor. You don’t knowwhat it is—and neither do I before I do some algebra!—but you say, well, if I callit x, I am confident it will have to go like so: The total volume, after adding in x,the correct amount of the 90% solution, will be 1 + x, one ounce plus the x ouncesadded. How much alcohol will be in there? Well, the alcohol can only get into thefinal mixture in two ways: either it was already in there or I added it. How muchwas in there at the start? Since it is a one ounce mixture 20% of which is alcohol,you get that 0.2 · 1 = 0.2 ounces of alcohol are in there to begin with. How muchgets added in? Well, of course, that is unknown since you don’t know how much ofthe 90% solution to add, but, again, you sense that there must be an answer—youare going on the hunch that such a (real) number exists!—and you just call it ‘x,’as a convenient pronoun. So, in terms of that as-yet-unknown value, anyway, theamount of alcohol added is 0.9 · x, meaning 90% of the total amount added. Thus,you imagine that if there is a solution, it will have to go like this:the desired 30% concentration of alcohol =amount of alcohol in there to begin with + amount of alcohol added intotal volume of the mixture at the end,or0.3 =0.2 + 0.9x1 + x.(d) Now you use the rules of deduction (which are all physically reasonable!) to deducewhat x must be. Here are the steps, for which you ought to try to supply a narrative:0.3 =0.2 + 0.9x1 + x0.3(1 + x) = 0.2 + 0.9x0.3 + 0.3x = 0.2 + 0.9x0.1 = 0.6xx