معرفی VICsفصل: بخش ریاضی / درس: مسائل کلمه / درس 22
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Intro to VICs
Some word problems present different variables in the prompt, and then all five answer choices are also in terms of these variables. So in other words there not numerical answers there are variable answers to the question. I will be referring to questions of this sort as VICS, where VICS stands for variables in the answer choices.
Many students find these questions particularly challenging. Here’s a simple example. We’re not gonna solve this yet, but just to give you a feel of this question. Jennifer can by watches at a price of B dollars per watch, which she marks up by a certain percent before selling. If she makes a total profit of T by selling N watches, then in terms of B, T, and N, what is the percent of the markup from her buy price to her sell price?
And notice all five answer choices involve the variables B, T, and N. And, of course, there are also 100’s floating around that doesn’t make sense because it’s a percent that somehow 100 could be involved. But this is an example of a problem with variables in the answer choices. So, how do we approach these? Well, in general we have two different approaches to these.
We can use either a straightforward algebraic approach or we can pick numbers for the variables. Both are perfectly valid, each with its own disadvantages and advantages. The principal disadvantage of a straightforward algebraic approach is that some students find it confusing or hard to begin. So, if you see how to do the algebra, by all means do it.
So when you do the algebra, that is a guarantee that if we do the algebra correctly, you’re gonna arrive at one unambiguous right answer. And so that’s the big advantage of the algebra. That it gets you to the answer right away with no ambiguity. The problem with the algebra is if you don’t know how to begin or you find the algebra confusing, that’s a drawback of the algebra.
The principal advantage of picking numbers is that it almost always vastly simplifies the problem. Often the entire problem becomes much clearer to folks when actual numbers are used. So it makes things very easier. One can often eliminate some answers by picking numbers but a well designed question will have incorrect answer choices that work for at least some of the more obvious selections or picked numbers, and we’ll talk more about this in one of the coming videos.
Each round of picking numbers maybe relatively quick but it’s hard to know how many rounds will be required to eliminate all four answer choices. So you might say that the algebra approach is a little more difficult, but it’s efficient and gets you to an answer quickly. The picking number approach definitely is easier, makes things much easier, but the question is, how efficient will it be, how long will it take to narrow things down to one answer.
So we’re gonna show examples with this question. So we’d already seen this question. Pause the video and see if you can solve this question and then we’ll talk about it. Okay, let’s talk about this question. First the algebraic approach.
B is the by price, and call S the cell price, which is not one of our variables. The, the profit per watch is S minus B, and for N watches that would be N times S minus B. That would be the total profit clearly. We can divide that to get the difference, S minus B equals T divided by N. Well the percent increase would be S minus B over B times 100.
And S of course is not one of our variables, but we can replace S minus B with what we just found that it equals. S minus B equals T over N. So put T over N in there, and then just bring N down to the denominator, 100T over NB. That is the percent increase.
And this is answer choice A. So that is an example of using a very straightforward algebraic approach. We introduced an extra variable, we fought through the whole thing in terms of variables, and we got to the answer right away. Now here’s a plug-in approach to the same problem. Exact same problem.
We’re going to solve in a completely different way. Let’s say that the original buy price per watch is $10. Let’s make the mark-up 30%. So that’s gonna be the actual answer to the question. A 30% mark-up. That makes the sell price $13, and the profit per watch is $3.
If she sells 20 watches, she makes a profit of $60. So, there I just picked numbers. Now how did I come up with those exact numbers? We’ll talk a little more about this in the picking numbers video coming up a couple videos from now, but right now let’s just go with these numbers. This means that if we plug in B equals 10, N equals 20, and T equals 60 into all the answer choices, the correct answer will yield a value of 30.
So now we have to plug this in to all five answer choices. So those, that’s what we’re gonna plug in. 30 is what we’re trying to get. Here are all the answer choices. We’re gonna plug this in. So start with A.
Plug this in to A. We get 100 times 60 divided by 10 times 20. Well, 100 divided by 10 is 10, 60 divided by 20 is 3, 10 times 3 is 30. So this equals 30, so this is a valid possibility for the answer. This actually works at least when we plug this in. It doesn’t guarantee that this is the right answer.
This just happens to be one answer that works for this combination of numbers. We plug in to B. Here we get 600 divided by 100, that would be 6. So, 6 divided by 20, that’s not going to equal to, so that doesn’t work. For C we plug in, we get 100 times 60 times 20 over 10. Well, 100 times 60 is 6000, 20 over 10 is 2, 6,000 times 2 is 12,000, this does not equal 30, this doesn’t work.
Choice D, in that parenthesis, 3 minus 10, we get the 3 from 60 divided by 20. 3 minus 10, that’s negative, so that’s going to be a negative number, that doesn’t work. And then E, when we look at this. We get 60 minus 200.
That’s also going to be a negative number, that doesn’t work. So, as it turns out for this particular choice. We eliminated four answer choices. We got very, very lucky with this choice of numbers. And so this allows us in, just one round of picking numbers, to eliminate four answers and get down to something, a single answer that works.
Here we see again that A has to be the answer. That problem was relatively easy, in that most reasonable choices of numbers, would eliminate the four incorrect answer choices, all at once. Harder questions are designed so that this is not so easy. If there’s any obvious choice for the numbers picked the incorrect answer choices will be designed to work specifically for those obvious choices.
So we’ll plug in and we’ll find that maybe two or three of the answer choices still work. So we won’t be able to eliminate four answer choices all on one go. We’ll eliminate two on one choice, maybe we’ll have to make another choice and eliminate the others, that sort of thing. We’ll see more examples of this in the lesson on picking numbers, coming up a couple lessons from now.
Here’s another practice problem, a variable in the answer choice problem. Pause the video and then we’ll talk about this. At the store, Sam bought a shirt and a toaster. There was an 8% sales tax on each item, and with tax, Sam paid a total of Q. So that’s the total amount he paid, the price of the items and the 8% tax. If the price of the toaster before tax was T, what, in terms of Q and T, is the price of the shirt?
Well, a few things I’ll point out about this problem. We can begin by eliminating answers that play on a common percent fallacy. Let S be the shirt price, T be the toaster price before taxes. The bill before taxes is S plus T. With an 8% sales tax this is Q equals the multiplier for an 8% tax that’s 1.08 times S plus T.
So that’s the correct relationship. To solve for T, we have to undo that 8% increase. As we’ve learned in the module on percents, an 8% decrease does not undo an 8% increase. If we increase by 8% then decrease by 8%, we do not get back to where we started from.
This is one of the principle fallacies of percents. And if this is unfamiliar, I would urge you to go back and watch those videos in the module on percent so you get clear on this. The answer choices with 0.92, the multiplier for an 8% decrease, are all incorrect. Because the correct answer is not at all going to involve an 8% decrease.
We want to undo an 8% increase, but that does not involve an 8% decrease. So anything where 0.92 appears is wrong. We can immediately limit A, B, and C. And in fact, once we have that percent equation, this is very easy to do algebraically. All we have to do is just solve for S.
So we’re gonna divide by 1.08, then subtract T, and that’s our answer. This is answer choice E. So this is one that is particularly easy to do with algebra. And in fact, if we see the percent error, that allows us to eliminate answers right away. So some word problems are VICs, they have variables in the answer choices.
For a VIC problem, we can pursue an algebraic solution, which may be harder, but will lead clearly to one answer choice. We can also solve by picking numbers for the variables, and use the process of elimination. And then the next couple videos I’ll be comparing those two approaches.
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