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Turning Variables into Numbers, Part II
This is episode two about turning variables into numbers, you should start this one if you're sure you really understand the basics and ready to build on it with this episode. First of all let's start out with a recap of what the last episode was all about. Turning variables into numbers has three general steps, first of all you assign numbers to your variables, then based on those numbers you solve the problem and lastly you look for the match between your answer choices and the answer that you found. Now we're going to add two more wrinkles to turning variables into numbers. First of all if answer choices include variables which we haven't seen so far, so far we've only seen answer choices that had numbers, then you're going to turn those variables into numbers as well. Secondly if more than one answer choice works then you're going to need to go back, change your numbers and go through the process again until you only have one answer choice that continues to work. Let's look at all this in more details using some examples.
So here we have an example of the first new principle which is that sometimes you have to turn variables in the answer choices into numbers as well, but we're still going to go through the same three basic steps so let's do that. Let's start by reading the problem now, 'if a, b and c are integers greater than one with ab=15 and bc=33, which of the following must be true?' So the first step is to choose numbers that work and remember you have to meet all the restrictions in the problems so choose carefully. Now you might naturally say okay if A times B is 15 maybe we'll try three for A and five for B. But one thing that makes this one tricky is that since this B is five, this B also has to be five and yet there's no integer for C that would work, five times nothing is going to give you 33, at least not given that C is an integer. So sometimes you have to back up and try again so this three and five combination must be wrong, so let's start over. Instead of three times five, let's try five times three that will fix our problem. We still have five times three as fifteen so that's good and yet now that B is three, we're going to be able to make this statement true. Three times what is 33? Three times 11. So we've beat all the restrictions in the problem and we're ready to solve the question. The question is which of the following must be true? So actually we kind of have to look at the answer choices in order to solve the problem and that's where we get to our new guideline which is that sometimes variables have to be turned into numbers. For instance to know whether B is greater than A is greater than C is true, got to put the numbers in. So let's do that what was B? B was three, so three is greater than A, that was five is greater than C which was 11, not so much three is not greater than either of those, so we'll get rid of that one. B; A is greater than C is greater than B, A is five is greater than C which is 11, we can actually quit here 'cause five is not greater than 11. C; C is greater than A that's 11 which is greater than A which is five and is that greater than B which is three? Yes. We've found our answer so we can quit now. So the principle here is that sometimes you have to take your answer choices and also make them numerical, if they're not already numerical. Let's look at one more example of that idea.
'If Q=rs which of the following must be equal to qs?' So yet again we need to pick some numbers and here's a little tip, if we pick Q first it might be a little harder to find R and S that are easy to deal with so we should pick R and S first and then have that determine what Q has to be. Also in general we want to avoid the numbers one and two, so let's try three for R and four for S and then three times four that makes Q 12. So we've already picked numbers and we're ready to do step two, answering the question. The question is basically what does QS equal so let's do that. Q is 12 and S is four so what is QS equal? 48 so that's our answer and we're ready to look for a match among the choices but wait they're variables not numbers so we need to turn them into numbers before we can find the match. So let's do all the plugging in. R squared times S is three squared times four that's nine times four which is 36 and that's not a match for 48. B; Q squared over R Q squared is 12 squared that's 144 divided by R which is three and that does turn out to be 48. Now it's up to you at this point whether you think that you want to quit because you usually will not find more than one match but just to be safe you should generally keep on going. So we tentatively think that this is our answer but to be safe, let's do the rest. C is R over Q that's three twelfths that's a very small fraction nowhere close to 48, so we're done. D is R squared over S that's three squared over four that's also a relatively small fraction, nowhere near forty eight it's out and S squared over R that is four squared which is 16 over R which is three, still a relatively small number definitely not 48. And so we've confirmed that the right answer is indeed B.
So there's that strategy but let's look at the other one right? That was that sometimes you're going to find more than one correct answer and remember when that happens you go back and change your answers and you keep on doing that until you've narrowed it down to one answer choice that is always right no matter what numbers you choose. So let's try this one; 'if zero is less than x is less than 1 and -1 is less than y is less than zero which of the following must be true? So let's start by choosing a value, let's try X equals a quarter and Y equals negative, let's see one half. Okay so is it true that X squared is greater than Y squared? Well this is going to square to one sixteenth this is going to square to one forth, one sixteenth is smaller so this is out sounds good to me. Now what about X minus Y is greater than zero? One quarter minus negative one half is one quarter plus a half which ends up being three quarters, is three quarters greater than zero? Yes it is so we're going to have to leave B in, it might be the right answer or it might just work with these particular numbers. Let's try C; Y minus X is greater than zero so that would be negative one half minus a quarter that's negative three quarters that's not greater than zero, so we eliminate that one. D, x+y is greater than zero, that's one quarter plus negative one half that's negative a quarter not greater than zero, so that's out. Looks like we're doing pretty good let's see how E treats us x+y is less than zero, well we just did that one right? And it was negative a quarter, so negative a quarter is less than zero also true, so we actually have it narrowed down to two choices.
So we don't actually know yet what the right answer is but we can pick new values for X and Y and again see which of the two answer choices continues to work no matter what. So let's mix it up a little before the size of X was smaller than the size of Y, technically I should be using the word magnitude. So let's mix that up, let's try making X into one half, still positive 'cause it has to be and making Y negative one quarter still a negative 'cause it has to be and then let's see how that impacts B and E. This time x-y is one half minus negative a quarter that ends up being three quarters and it's greater than zero sounds good, still true. Down here let's see if this looks true x+y is less than zero, one half plus negative a quarter well that's one quarter less than zero? No longer true, so we've managed to eliminate E, B remains true so we've found our answer. There's more than one correct answer at first but by changing numbers we were able to narrow it down to just one right answer.
Let's look at one more example of this idea, 'if -1 is less than n is less than 1 and N is not equal to zero which of the following must be true? So let's pick a number let's try N equals a half. We do have to make sure that we're between negative one and one, a lot of people naturally pick integers so we'll try one or two or three but that clearly violates the rule so watch out for that. Now let's see what's true is it true that negative one half is less than one half? Yeah that works so we leave A in. Is it true that one half is less than one half squared which is a quarter? Nope one half is actually bigger so B is out. Is it true that these are just reversed is it true that one quarter is less than one half? That's true so we're going to leave C in as an option. N cubed that's going to one eighth, is it true that one eighth is less than N squared that's one half squared which is a quarter, yap an eighth is less than a quarter so we'll leave that in. And E is it true that one half is less than one half cubed which is one eighth, nope that's out too.
So as you can still see we still have quite a bit of work to do. Let's mix it up and this time since we tried a positive fraction let's try a negative fraction we want a different kind of number. But to keep it simple we can stick with negative one half 'cause we know how to deal with one halves efficiently. Looking at A, negative N is now equal to one half and is that less than N, negative one half? No A is out now and I want to point out that we often think of negative signs as meaning that we're dealing with a negative number but that's not true. In A, with this value of N that negative N just means switch the sign so instead of meaning it's a negative number it means it becomes a positive number kind of unexpected. C; N squared is negative one half squared and that's a quarter is that less than N which is negative one half? No a negative number is not bigger it's actually smaller so C is out. So it looks like D is the right answer but let's just verify N cubed is negative one eighth because remember negative numbers