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Use Your Figures

Teacher/Instructor Eva Holtz
Eva Holtz

Harvard University
Perfect scores on the SAT and 4 SATIIs

Eva is a certified admissions counselor and the founder of PrepPoint, a premier test prep company in the San Francisco Bay Area.

In this episode we're going to talk about how best to use your figures on the Math section. Now it's ideal if you can do the real Math but this will come in handy sometimes when you're running low on time or actually you did the real Math but want to see if you're answer made sense, or if you're feeling stuck and want to make an educated guess. So let's have a look at how you can use your figures. First off there are two kinds of figures, they are inaccurate figures and they're always going to be labeled so you will know when you see them because it says note, figure not drawn to scale. When you get inaccurate figures you can just consider redrawing them and then using your redrawn figure to draw inferences from. And then on the other hand we have accurate figures they aren't labeled, so at first it might be confusing but just get used to the idea that if a figure doesn't say it's not draw to scale you automatically know that it's drawn to scale. And when you have it figured like that you can use it to estimate things like angles, area and length. Let's have a look at a few examples so you can see this in action.
So here we are with an example of estimating angles, now like I said ideally you would do this using real math and I recommend that. But let's say you're stuck, you're running out of time or you just want to check and see if your answer makes sense then you can estimate. So first off we have a diagram and it doesn't say note figure not drawn to scale, so we automatically know it is drawn to scale and that means what we see is what we get. So the question says what's the value of X? In other words how big are these angles? So we can estimate right off the diagram if we have to. Now one way to do this is to compare X to 90 degrees. So I'm going to draw a 90 degree angle right here then I'm going to split it in half 'cause I know what 45 is like, half that. And then if I look at this I'm like well it looks like I could fit three Xs into that 45 degree angle. So that must mean that each X is 15 degrees that's my best guess. Now the right answer might be a little different from that but I think it's right around 15. And sure enough, if I look at the answer choices 15 is an answer choice so that sounds really appealing, 20 is also an answer choice and that's pretty close so I might also consider that, but C, D and E seem pretty far off given that I know that my figure is drawn to scale. So at this point I can make an educated guess and I may not necessarily get it right but I hope you remember that if you can eliminate one answer choice you should guess. So either way it's worth guessing and my best guess is A, 15 that also turns out to be the right answer.
Let's look at another example of one of these where we estimate angles. This one is a little different because the figure is not drawn to scale. So what that means is we're going to want consider redrawing it so that we can draw conclusions based on an accurate diagram instead of this inaccurate diagram. So first off I need length L and M to be parallel which they're not quite in the original. And then I need this angle here to be 75 which is not quite in the original looks pretty down close to 90 degrees there. I'm going to slant it a little more so this is 75 degrees now it's more accurate and I want to make this so that this is Y and this is two Y in other words this angle should be twice as big, which is definitely not happening in the original. So I want to break this angle up so that there are two parts here and one part here. So I'm going to do my best to do that and this is what you should do in your text booklet. That doesn't look quite right so let me try that one again. Let's try a little more like two to one let's see a little more like this.
Okay now it looks a little more like, I could fit two angles here and one angle there it's a little better. Alright so the question is what's the value of Y? Well let's see a 90 degree angle would be about this big and a 45 degree angle would be about half that, this big. So it looks like the value of Y is a little less than 45. So let's see is 15 a little less than 45? No that's a lot less than 45 same as to have 25, 30 is pretty close, 35 is pretty close and then 75 doesn't sound good at all. Yet again just by making an educated guess based on a well drawn diagram we're down to two. The right answer happens to be D 35 so not bad a 50-50 guess based I'm not doing any real Math.
Now the second thing we can do is estimate area, so here's an example where we estimate area. We're told in this case that we have a circle with an area of 16 pie. Now if you're feeling totally stuck here's what you can do. You can say the area of a circle is 16 pie and take that 16 and multiple it by pie which is roughly three point one four, it ends up being if you use your calculator or do some straight up old school multiplication 50 point two four. So here's what you can do if you can't do it "the right way". You can say okay, so I know my circle has an area of a little over 50 so what do I think the area of the square is? It looks a little less but not a ton less, it's not half or a third or a quarter. So compared to 50 we expect it to be a little smaller of an area for the square. So 64 is way off that's actually a bigger area, 32 is promising 'cause it's a little smaller than 50 and 16, eight and four don't sound right at all. And so in this case without doing the Math we've actually narrowed it down to the right answer which is pretty cool.
Let's look at a similar example where we estimate area but we actually have to redraw the diagram. Here we go. We know we have to redraw because we're told the figures are not drawn to scale and we can see they're not drawn to scale for a couple of reasons. First of all this radius should be twice this radius 'cause it's two R versus R and that appears not to be true in the diagram. Also this should be three N versus N, should be three times as big and it appears not to be. So let's redraw this as best as we can. We'll have one with a radius of R and one with a radius twice as big, let's see does that look about twice as big? Yeah. So now based on that let's draw a circle, this could talk some doing it's not easy, so if you have to re-erase and redraw that's fair enough. This one it's going to be about... like this, not the pretties thing you've ever seen but I hope you'll bear with me on this one. And then we have to make the arc have a larger angle three times as large. So let's make this N and let's make this over here three times as big, about that, one, two, three okay. So the question is how many times larger is this shaded area than this shaded area? And just by eye balling it I think you can tell a lot of this will fit in, let's kind of see how many. It's a little time consuming but this would be one of the harder problems on the test so you'd only be here if you were already pretty strong in Math and pretty quick at Math.
So let's see if we can fit one in here, maybe another one in here, that looks like we could fit about one, two, three, four so far, maybe another one here five, six, seven, eight, nine maybe ten, eleven. It's a little ugly but we're saying okay I think I can fit about eleven of those. Let's see what looks like a good match. Now eleven isn't an answer choice and that makes sense we're just doing an estimate and God knows it's not pretty, but you can tell that there aren't five the little ones in the big one. There aren't six, twelve sounds pretty promising, eighteen that's a stretch, the difference between 11 and 18 is pretty pronounced. And 36 definitely not we cannot fit 36 of these into here. So sure enough without doing the real Math we've got the right answer C there are 12 of these in here.
Now we can also use this approach for estimating length. Ideally you would remember, this is asking a lot, that any triangle that is inscribed in a circle is automatically a right triangle if it goes through the center. If that's the case, you would be able to apply the Pythagoras theorem and find out that this length is ten. And that would be great but let's say you didn't remember that 'cause a lot of people don't and may have been a while since you took geometry and that's not one of the most common pieces knowledge from geometry. In that case you can still use length estimation to solve the problem, let me show you what I mean. You could literally take your pencil and line up your pencil with the diagram and put your finger nail where the diagram ends so you could measure this length. Now I don't have a pencil this long, so I'm going to use my hands but of course on the test you'd use your pencil and you figure out how long it is.
So this is eight and then I swing it around over here and I say okay, if from here to here is about eight what do I think that length is? My best guess would be maybe nine and a half or ten and then I could use that to calculate the circumference. I could use if I think it's ten, use the formula circumference equals pie times diameter and that equals, well pie is pie and diameter is I think about ten, so this is my best guess for the circumference even if I wasn't able to find out for sure this is ten just approximately ten. And I look at the answer choices and sure enough there's ten pie and five pie is not close enough but I think it's a good option, 14 pie is not close enough but I think it's a good option and definitely D and E are just way too big. So yet again I've got it down to one right answer which is pretty awesome but even if you can get it down to four or three or two it's still worth guessing 'cause you should always guess when you can eliminate one answer choice or of course more is always better.
Let's look at one more of these. Now in this diagram we have to deal with something that's not drawn to scale so again we have to redraw it. So it is drawn to scale and then we can draw our conclusions. So let's see this piece looks pretty good because it's a 45-45 90 triangle, it makes sense the two lengths should look the same and it does look like they're drawn to be the same. So we're happy with that piece, it's this bit over here that isn't drawn to scale. This is supposed to be a 30, 60, 90 in order to add up to 180 degrees but these angles look really close to being the same, not as different as 30 and 60 should. So let's redraw over here this first triangle going to make pretty much the same, but the second triangle I want to make that angle bigger, so let's see. I want to make it about 30, 60, 90, not the prettiest thing you've ever seen but that's close, this angle does look about twice as big as this angle so it's more reliable of a diagram now.
So the question is if this guy is root two, how big is this guy? So let's start by turning our numbers into decimals 'cause you probably can't estimate very well with roots but decimals you can handle. So one point two, I'm sorry, so root two happens to be one point four, more or less and the question is if this is one point four how long do we think this is? So let's see if this is one point four and of course you do this with your pencil not your entire fist, this is one point four then we'd say okay I think this is one point four up to here and I think this is one point four up to here. There's a little left over maybe a half, so maybe it's about one point four plus one point four that's two point eight plus a little bit maybe three point six or so that's our best guess, something like that. We don't know for sure but it's an estimate. Let's see what these are worth. Two, two is pretty far off than three point six so we don't think that's a good estimate, two root two happens to be two point eight so that sounds a little low we can live it in there and come back to it but it's not very promising. Two root three that ends up being three point five, that's closer. D is four, four is four that's easy, so that's pretty promising too and four root two that's a pretty big number, that's five point seven so that's definitely out. So it looks like what's closest to three point six is one of these guys and probably not this guy. It happens to be the case the right answer is D, you might have ended up going with C but 50-50 is not odd so if you'll notice that we got most of them right and one of them wrong, overall a pretty good effect. But it does go to show that real Math is better than fake Math but this is good in a pinch. So let's summarize everything we've covered.
So on the SAT Math you'll have a lot of figures to deal with and some of them will be inaccurate and you want to remember with those you should not...