AMC 8 Math Enrichment Program Lect 9 How to Solve AMC Problems in Seconds

All right.

>> Yeah. So that's the topic of today. I

mean last last time we also did a little bit right.

>> So how to solve AMCA problems in seconds.

>> Good.

>> And then and then we're going to also do some geometry problems. >> Especially when I look at it in a different direction.

>> Really? Oh yeah. I remember last time >> you're looking at through >> the answer. I almost got authoritis.

>> Oh, okay.

>> The best part.

>> All right. So, let's take a look at this problem.

>> Like if it's simple, can you do it in 10 seconds? Right.

seconds? Right.

>> I have to read it first. I have left.

>> He answered 204 kilome kilometers.

>> Very good.

>> I did it because like is the only logical answer.

>> All the other answers are a third.

>> Yeah.

>> Yeah.

Yeah.

>> Okay. So

the smarter of you already figure out.

So basically it has to be proportional, right? So this over that is equal to the

right? So this over that is equal to the new number over so we can put it this way

should be the new number over 34 and I mean just by looking at it so this is roughly 50% larger so so the number

should be roughly 50% larger >> yeah like >> so that that leaves out you know the only option is eight but you can also calculate it I mean this is just six

right you can do six times >> 34 okay >> because the other either is too small too big okay

>> that's how you can do it in in a few seconds >> oh you cannot see what I drew on the board I'm sorry that's okay Um

all right.

So this one >> right attention please. Let's do it.

Is the answer 15 B?

>> We got 15.

>> That's good. Yeah.

>> Okay. So, some of you have mastered the way to do it in a few seconds. That's

that's awesome.

>> And and others are still reading.

>> Okay.

>> Yeah. So, this is basically um it's probably too simple. Are people

still consinging?

Okay, got it. So, um, yeah. So, you

basically choose, you know, six out four out of six, right? So, that's the same as choosing two out of six.

Um, all right. So, it's C 62 or C6 four.

It's 15. And

next one.

Wait, >> I remember needing some time to do this.

>> Okay.

>> Say the answer is 57.

>> Is the answer 59?

>> Mhm.

>> It's an odd number.

>> Yeah, it should be an odd number.

>> Yeah, 59. No, it's not. 57. I don't

know. Do

>> you agree?

>> Yes.

57 or 59 forget 59. But like I also have really

forget 59. But like I also have really bad money so I should run.

>> Okay.

So it's basically for one team to be the final winner like how many needs to be eliminated?

Right? So it's the same question as what is 60 minus one?

Can you finish that in seconds? Yeah.

>> So, of course, it's D, right?

So, so it's basically boils down to how you translate the the problem description into you know mathematical equations or expressions then you can

finish it. Okay. Now this one

finish it. Okay. Now this one let's not remember anything. Let's let's

forget it and just do it as a new problem. Okay. And see how fast you can

problem. Okay. And see how fast you can get it right.

>> Again accuracy is first priority and then speed.

It's E.

It's E. What?

>> That's Yes. E.

>> It's E.

>> It is E. I just

>> All right. Let's give other people a little more time and then we'll talk about it.

Okay. So, Charlene, how did you do it?

>> Uh, I just like added all the >> addit all the times.

>> Yeah.

>> Okay.

>> 1.5 hours.

>> So, so you could do 1.5 hours plus 90 minutes is also 1.5.

And then this is basically you know it's lacking 10 minutes to become four hours but then we have another 10 minutes so it's >> multiple of 11

>> exactly so you could do this and then times 11 but you could also observe right because this 11 and there's only one multiple of

11. So this would be the fastest way

11. So this would be the fastest way right you don't even have to do these translations. So so if you can you know

translations. So so if you can you know figure out what is the key in this problem like key features then you can solve it in seconds.

>> You could have worked five and a half out.

>> H >> could have worked five and a half hours.

>> Five and a half.

>> Yeah but all these are yeah integers. So it's

like family >> and all these are integers. So there

must be a multiple of 11 and there's only one choice.

>> Okay. So this is an example. You know

there are many different way to solve the same problem. But in MC8 speed is very important. So that's why making

very important. So that's why making observations and eliminate uh you know impossible results will will give you much quicker way to solve it.

Does that make sense? online people.

>> We have three people online.

>> Well, because they're not in the state.

>> Oh, >> not not in New Jersey.

>> But there's people that >> I live in California.

>> Living in Michigan.

>> Yeah, I think they're in California. All

right.

>> Are you not? All right, let's >> cal for him.

>> Yeah. Okay. All right. Next. Let's work

on next.

>> How do you know the times?

>> How did he know to come here?

>> Okay.

>> Yeah. How did find out about this place?

All the way in California.

>> Well, how does he know about this tiny place?

Well, >> her mom contacted me.

Oh yeah, my mom knows this place because she would got she got added into a random.

>> Let's focus on the math. Okay, we can discuss that later.

>> Oh yeah, I remember doing this and getting extremely confused.

>> Everything I remember getting this one wrong.

>> Okay, just do it again. Do has a new problem. Okay,

problem. Okay, let's see.

Watch me.

>> Is that B?

>> Well, calculate it. Don't Don't try to recall anything. Okay, let's assume it's

recall anything. Okay, let's assume it's a new problem.

>> How do you calculate it? So,

>> in the middle, >> it could also be D. It could also be E.

>> It's definitely not C.

>> It's definitely >> definitely not C. Yeah, I agree.

>> It's like way too far. Then

>> the maximum it can possible last and then >> so for this you probably have to calculate it.

>> It's never mind. I have a paper. I don't have

never mind. I have a paper. I don't have it out though. I don't know why I got a notebook out but not pencils.

>> Yeah, it helps you for you to keep notebooks. I think you know in the

notebooks. I think you know in the future you can also review what you learned in class, right? Otherwise you

you come and then you learn something you may just forget about >> if they answer um um B.

>> Yeah. See

>> Ethan Long.

>> Huh?

>> How old is he?

>> How old is Victor? I think he's eight.

Yeah.

>> Yeah. GPT.

>> So I I I saw it like it's kind of joke on on the forum, internet forum. It says

like AMC8 is for eight years old and 10 is for 10 years old.

>> Can I do the AMC 12?

>> Yeah. When you are 12. Yeah.

>> Wait. Why isn't he going to >> like the numbers?

>> He's in third grade.

>> No, no. AMC8 is for eighth grader or below. But yeah, that's a joke. Yeah,

below. But yeah, that's a joke. Yeah,

cuz a lot of lot of eight years old attend AMC8 and >> let's focus on the let's focus on math.

Okay.

There are different ways to do it. No,

if if you have figured out a way like think about are there any other ways you can obtain the results. Okay, that

helps.

Yeah, I don't know how you get 47.

>> Um, this is a speculation, but it's just because >> well 120 is half of 240. Then 15 - 2 is

13 and 45 - 13 is tw is 32. And then 32 plus the 15 again becomes 47.

>> Okay, you found a way to make a 47 out of it.

>> Well, it has to make sense.

You're like playing the game of 24.

Is >> he in his go to like >> Why Why is he not in a California?

>> Why is he only eight in this?

Victor.

>> Okay. Should we look at this together?

>> Okay. So, let's make some observations.

Okay. So, the uh Oh, maybe Victor, can you explain how I did it?

>> Victor, are you there?

So, I took the 15 over 120 is how much she's using it constantly over how much it will last like if using it.

>> And then over 120.

>> Yeah.

>> Yeah. And that represents >> that represents how much like she was using it over like when she's using it over he's not using it.

>> Yeah. The ratio between using and not using. Okay. And then since 240 minutes

using. Okay. And then since 240 minutes is just 4 hours 15.

>> So this is 4 hours. Yeah.

>> So then 15 - 4 is 11 over x because that's how much because 240 like it that's pretty much how much she was

using it. And then x is how much like

using it. And then x is how much like over she wasn't using it. So 88

and then so 45 - 41.

So, so because that's how much um we didn't use it. So then 88 minus 41 is 47.

>> Yeah. Very good. Uh did you follow guys?

>> Oh, okay.

>> Oh, I'm not writing. You're not you're not seeing what I wrote. I'm sorry.

>> Let me switch. Yeah. Um

>> he probably couldn't hear you, but you could you hear him? So, okay. So, Victor

was doing >> All right. Let me write it down here for you. Okay, so 15ide by 120. What does

you. Okay, so 15ide by 120. What does

that represent?

>> Wait, what are you doing?

>> Using it, not using it >> over.

>> Yeah. So that's that's basically >> 60.

>> The the ratio between using it versus not using it. And um this is four hours.

So he used it for 4 hours 20 240 minutes, right? So that means there's

minutes, right? So that means there's equivalence of 15 minus 4 left right. So so during those four hours it

right. So so during those four hours it loses four loes four hours of using time. So 11 hours left and that is uh

time. So 11 hours left and that is uh divided by x. Right? So here this is basically one over eight. So x equals 88. Oh,

88. Oh, >> right >> now 88 is like basically basically we have

88 hours of not using time left. That

make sense? Now how many hours of you not using time we already used? So that

is 45 hours minus 4 is 41 hours not using time.

Correct?

So we have 88 hours of no using time left in any um yeah and subtract that 41 and that's how you get 47

>> crime number that that's how uh Victor did it.

Um and there are other ways to do it. Are

there any questions regarding this approach?

All right. Very good.

I think my solution was >> okay. Um, yeah, that's exactly how

>> okay. Um, yeah, that's exactly how Victor did it.

And then Okay, another way. So, the way I did it was basically let me clear this.

So, I use fractions. You You guys have learned fractions.

Okay. So what I did was let me erase this.

Can I erase everything?

Yeah.

So what I was saying is thinking is um each hour like not use when when she's not using it, it

lasts for 180 120 hours. That means each hour not using it, it costs one over 120.

That make sense?

And if she's constantly using it, the battery lasts 15 hours. That means each hour of constantly using it, it loses one over 15.

Make sense?

And then how how long has he has she used it? So she has used it for 45 hours

used it? So she has used it for 45 hours among which there's four hours of constant using it. So basically the

usage was 41 times 1 over 120 that's you know when she's not using it.

So so out of the 45 hours there are 41 hours of not using it not constantly using it and there are four hours of constantly using it.

Right? For constantly using it the consuming rate is one over 50.

And so um so this is basically how much she has used and um and from here if she doesn't take any so how many more hours

battery lasts. So that's basically one

battery lasts. So that's basically one minus this. It's getting a little bit

minus this. It's getting a little bit complicated and then divided by the consuming rate. The consuming rate is

consuming rate. The consuming rate is one over 120.

So that's another way you can do it through fractions.

is not nearly as elegant but it it takes less the brain power. So basically you just write down what is the consuming rate for not constantly using it. What

is the consuming rate for constantly using it? And so how much has been used

using it? And so how much has been used at this rate, right? That's 41 hours and you know four hours of constantly using

it. Then we want to find out how much

it. Then we want to find out how much power is left. The what fraction of the power left is one minus that and then divided by the consuming rate of you

know not constantly using it.

Okay.

So again, there are many different ways to do it and you know, you just have to figure out what fits you the most and and it's nice to know multiple ways.

That way, you know, if you have extra time, you can use a different way to check the answer.

All righty.

Okay, let's take a look at this problem.

to the >> Oh, yeah. I remember getting this one wrong even though I got it right.

>> It's C.

>> It's E.

>> Yeah, it's C.

>> So, it's C.

>> It's a E.

>> Okay. Double check the answer. Double

check the answer. If you pick one, see if it fits all the, you know, description.

No, >> but only one of the statement is true.

>> No, it has to be >> check. Double check. Okay, once you pick

>> check. Double check. Okay, once you pick the answer, guys, once you pick the answer, check the description whether it's fulfilled.

>> Statements is true.

>> Yeah.

>> Bob is the oldest. If Bob is >> So Bob is the last one.

>> Alex is not the oldest.

>> Yeah, >> it's youngest oldest.

>> Oh, no, no, no. He's right. He's right.

>> What is true? One can be true is the oldest. That means Alex is not the

oldest. That means Alex is not the oldest is false. That means Alex.

>> That means that it has to be right.

>> C is right. If all the word >> Wait, it has to be B. No,

>> Kathy is wait. No, Kathy is not the oldest. Sorry.

oldest. Sorry.

>> Okay, so >> Okay, guys. If

if Bob is the oldest is true, then the second must be false. Alex is not the oldest, which means Alex must be the oldest.

>> What? What?

>> And it's in conflict. Does that make sense?

>> So if if number one is true, then Bob is the oldest. Then number two must be

the oldest. Then number two must be false, which means Alex must be the oldest, >> which is in conflict with

above being the oldest.

That make sense? So that means one must be false.

Make sense?

>> Which one then is D?

>> Does that make sense?

>> I think it was E.

>> You have to understand the logic reasoning.

>> B.

>> Oh, it's E.

>> It's E.

>> Well, double check. Once you pick the answer, double check whether all the criteria are fulfilled.

>> Not the it's a >> double check. Double check using all these criteria.

Yeah, that's right.

>> But Kathy is not the youngest.

>> Kathy is not the youngest.

>> Kathy Kathy is saying >> the third statement is true. So Kathy is not the youngest.

>> Yeah. So it's not >> and then Alex is not the oldest is false and Alex is the oldest and then Kathy >> No, but a not the oldest.

>> No, but if >> if Yeah. So but that statement's false.

>> Could be the youngest.

>> Then C. No, then that that statement doesn't really apply to any >> is it not?

>> I don't know.

>> Well, again, once you pick your answer, check all three criteria.

And is it >> is it only one of them being true?

>> Yeah, it's C.

>> Check. Check.

>> Yeah.

>> Whether only one of them is not the youngest.

>> Yeah, exactly.

>> Yeah, it is. Cathy's in the middle.

>> Said we said Kathy was C was the right one. entire time.

one. entire time.

>> But Bob is not >> Bob is the Well, we already ruled out Bob being the oldest. Remember the

number number one statement must be false, right? If the if the number one is true,

right? If the if the number one is true, then number two must be false, which will lead to Alex being the oldest.

Now, it's in conflict with Bob being the oldest.

>> Bob, the Bob one is true, then the Alex one is false. D.

>> Bob one cannot be true.

>> Wait, that's D.

>> It's D.

>> I'm still going back to my theory with >> Bob.

>> I got E.

>> Then Alex is the oldest. And we can >> Jod, what did you say?

>> If it's D, then Kathy is the youngest.

>> I got E.

>> Okay, we got >> Jodine got E.

>> And and Victor also got E.

>> Yeah, I think it's E.

>> Okay, Victor also got E. E and E are 25.

>> It's definitely not E though.

>> No, I got It's definitely not E.

>> Oh, it's in the middle.

>> You can check.

>> Every single statement has to be wrong.

>> Check C. Wait, every statement has to be wrong for D. Every statement is wrong.

So, it's E.

>> But I got >> Yeah, if you D, then every statement is wrong.

>> Check.

>> Let's try. Let's try.

It's E. It's been E this whole time.

Yeah.

>> All right. Then you

>> wait. No,

>> it's E.

>> Oh, this is not the >> Okay. So, I I think the

>> Okay. So, I I think the >> I think the easiest is like like we start from number one, right? If Bob is

the oldest, then number two must be false, which means Alex must be the oldest, which is in conflict with one.

And therefore our assumption that number one is true must be wrong. So that will immediately rule out the the first one being true. Does that make sense? And

being true. Does that make sense? And

now there are only two possibilities. Is

number two be true or number three be true?

So you can you can keep trying. So so

basically number one must be false and that we can already rule out C, right?

So Bob cannot be the oldest.

Um and then if number two is true, then Alex is not the oldest. Um

>> then then Alex and Bob are are both like the first two, but then Kathy has to be the third one. But Kathy is >> then Kathy in this case Kathy has to be

>> so number three must be if number two is true, number three must be false. That

means Kathy must be the youngest.

Right? So if Cassie is the youngest, let's see this this option and you will find out you know all the other two does not satisfy and therefore D should be

root out and this should also be root out. So number two must also be false.

out. So number two must also be false.

Okay now we we only have one option left is number three being true. So, and then you can find out when number three is

true and one is false and two is fourth.

So, when two is false, Alex must be the oldest, right? And now we only have one

oldest, right? And now we only have one option because D cannot be the right choice. There's only one option with

choice. There's only one option with Alex being the oldest.

And Alex just logged in. And Alex is well in our current classes. She's he's

the youngest.

I'm talking about Alex Lee.

>> He's online today.

>> Um he just lo >> the older brother.

>> He's the younger brother.

>> Okay guys, does does that make sense to you? Who's

you? Who's >> this type of logic reasoning? I hope you can master it.

>> Um, and then find try to find the the the quickest path to um finding the right answer. Okay? Because AMC 8 is

right answer. Okay? Because AMC 8 is only like 40 minutes, you have to finish 25 problems. So, if you can finish one problem huh?

>> 45 >> 40 problems. >> 40 minutes. 25 problems, right?

>> 25 problems. >> No, no, no. 25. 25 problems. Yeah.

>> What the heck?

>> Um >> 40 minutes or 40 minutes?

>> 40 minutes and 40 problems. >> 40 problems in 25 minutes in one.

>> Not doing that.

>> 100 minutes.

>> All right. Let's move on to this one.

>> Oh, yes. Yes.

>> Oh, you like this?

>> Oh, I think I got this one right.

Okay.

>> 21.5 >> cuz I remember getting 21.5.

>> Okay. How do you >> I just did. No, I found the arrow >> behind somewhere.

>> I see your rear end >> here. No rear end.

>> here. No rear end.

>> Yeah, but now I see your head.

>> I think I think I found like >> I'm going to do a split and break my black.

>> I didn't find the actual area of the cape. I think I found the area of the

cape. I think I found the area of the outside.

>> Yeah, same. It's easier.

>> Oh, okay.

>> What's the area of the outside?

>> Cuz you can like make a bunch a ton of triangles that >> shame.

>> I'm sorry.

>> One, two, three, four, five.

>> See that? This is like a picture taken from a book because it's like >> Wait.

>> Oh, yeah.

>> The answer is >> See, >> just pick the middle one. It's the one in the middle. See if there >> one. No, I'm being lo in the middle.

>> one. No, I'm being lo in the middle.

>> I'm pretty sure I got C.

>> No, I'm being lo 22.5 is close and they finally put 22.5 because it's one that notice that cheap.

So 21.5 >> Wait, wait. How do you do that?

>> Stupid, but >> okay.

One away from each other. So it's more likely to be one. But since there's a one fold, they're probably gonna try to trick people into thinking it's still a bigger one. So it's so so it can't be

bigger one. So it's so so it can't be 22. So it's probably gonna be 21.5

22. So it's probably gonna be 21.5 count.

>> Wow.

>> But but when there are two choices that have decimals, right, and three choices that have integers, but usually people tend to think that, you know, it's more

likely to be one of the three. But 22.5

and 28.5 don't look like 21 that's 30.

>> So okay Charlene's argument is that because there's a hole this error is one. So they must trick us by having two

one. So they must trick us by having two numbers two two answers that defer by one and therefore we can rule out a DE immediately.

>> And then and then because one of them should be minus one. So let's pick the smaller one.

>> So when you calculate the missing >> 42, >> they literally tell you to go lower.

>> There's noise 42 and it's E.

>> Okay.

>> See, >> I think it is.

>> Oh my god. I didn't

>> I focused. That's focus. Okay. There

again there are multiple ways to do this problem.

>> See which way you end up with.

>> Trust me bro.

I can see. I just can't count.

>> I can't see.

>> Okay, I got it inus. There we go.

>> What? I need to put my finger.

>> It's a 10 by 10.

>> Oh, no.

>> It's 11 by 11. 11.

>> It's 11 by 11.

>> It's 11.

>> What?

>> 6 7 8 9 10 11 >> 11 dots. There was 11 dots.

>> Oh yeah, >> it's 10 spaces for 11 dots.

>> 1 2 3 4 5 6 1 2 3 4 5 6 7 Oh my god, >> I can't count. Okay,

>> look. I got 21.5. I am beating a 10 by 10. Well, it's actually 21.5

10. Well, it's actually 21.5 >> technically my level, but I'm not educated. Guess what?

educated. Guess what?

>> 21.5.

>> I'm almost there. I'm almost there, people.

>> All right, then.

>> My brain is working.

>> I mean, you could have just like calculated this. You could have

calculated this. You could have >> You could just calculate the exterior like how would I prefer ours? I did

this. I did the exact same problem two days ago in my other class.

>> What other class?

>> RS now.

>> Find the area of the Nike logo, but someone strongly aring up the entire class.

>> Find the area of one of one by one.

Wait.

>> Just like send it to take a video.

>> You know, Chachi means he's not going to answer second question. chat when I # an addition problem. I got it wrong.

addition problem. I got it wrong.

>> You said 3.25 3.14129.

>> Why are you doing this to the Nike logo?

>> It's a language a large language model.

I just have a deck of what letters >> so >> no the year was the year 2025 year >> 34% >> I got

I got >> you see that's why I'm so I got first I know you're still >> I don't >> like 157.

I don't know the last one.

>> Yes, all of my viewers made it. This is

one of my actually I remember eating this and >> what what's the skin called >> but my viewers maybe >> I practiced

game you saw that game before it's like you join the game and all you do is just keep on dying and to leave the game you have to Lego box.

>> That's real.

>> I'm going to go down here and make Lego.

>> What is >> waiting on 6.5 >> 648?

Did you spend Robux under >> 70?

I'll help you out.

with eight.

>> It's what is it?

>> 1 2 3 5 6 7 8 9 >> There's 10 There's 10 by 10 spaces, but 11 by 11 dots.

>> No, they the dots don't matter.

>> Yeah, it's 100.

>> The dots you don't matter.

>> Is that the dots matter? the dots.

Where's the other matters?

>> Oh, yeah. That's 25, right?

>> Is that Wait, is that a five?

>> Is it B? Oh, that's a seven.

>> What?

>> No, it's B.

>> I think you forgot my >> I I I Oh, yes. This one. It's C. It's C.

It's C. Yeah, see. We knew it. We

already knew.

>> Yeah, I got it first. I got it first. We

already knew it was C. You knew that.

Like, is it C? Okay. How you got it?

Yeah. Here, show me.

>> So, I needed to find the exterior first.

>> Okay, good.

>> It's very easy to find the exteriors.

>> Okay, >> 20.

>> This is >> how do you find out this one?

>> Oh, I cut it into So, this is four. This

is two guaranteed. This would be approximately 4.5. I know that somewhere

approximately 4.5. I know that somewhere here there will be an answer to >> this one. Why didn't you calculate it?

10 times Oh my god.

>> Right behind you.

>> You didn't know that.

>> Okay.

>> Okay.

>> So, everyone need like the same build.

>> Yeah.

>> But I'm glad you got it.

>> That's so great.

>> This is how much go.

>> This is 10.5.

>> Look at YouTube. The first thing that pops up. I think so. Yes. Well, wait.

pops up. I think so. Yes. Well, wait.

Let me check.

>> What?

>> Too short.

>> What the 10? I got 10. Not 10.5.

10? I got 10. Not 10.5.

>> Okay. But are we Oh, yeah.

>> Okay. Who's at C? Victor. Yeah.

>> Good. Okay.

>> How do you calculate it?

I found the exterior is 10 * 10 is 100 and then I found that the outside of the check mark or whatever it is is well I

split it into tiny triangle and then I got like 1.5 17.5

I mean 21 and um 30 and 10 and then I subtracted one and then I got

21.5 is the answer right so okay very good so basically what

Victor did is the the area is okay so the total area is 100 right it's 10 by 10

and then subtract ract um this part which is 4x five right

and then this part this part is 4 * 5ide by two right

and then this part is um what is this part 6 * 10 /

that are you following me? So the the areas of triangles >> yes >> um two legs divide by two and then plus this part which what is this part?

This part is 6 * 5 to two.

Am I right?

And then there's the monop part.

>> Instead of having the four by five, I had a 3x5, but it's kind of the same thing.

>> Yeah. So, and then it's three * 1 to two. Okay.

two. Okay.

>> 76.5 minus.

>> So, that's that's one way of of doing it. Okay.

it. Okay.

>> And oh, you need plus one at the end.

>> 7.5.

>> Um, >> well, sorry.

So, so there's another plus one inside.

Okay.

>> So, excluding the area. Um, and there's another way of doing this type of problem which does not involve a lot of heavy calculations and that's the method

what I call I have B minus one. I have a dream.

>> It's called I have B minus one. Okay.

What does that mean? Um, so it's it's Pix's theorem.

Um it's basically saying the area of a polygon is the area is equal to so I I I tend to like it in this way. So I plus

half of B minus one. So I have B minus one >> plus.

>> So I half B and then minus one. Okay. So

what does that mean? So I means how many lattice points are inside the polygon

and then B is how many dots are on the boundary. B means boundary boundary of

boundary. B means boundary boundary of that polygon.

So basically we just need to calculate these two numbers. One is I means the interior. So how many lattice points are

interior. So how many lattice points are within the interior of a polygon?

And then B is how many are on the boundary of a polygon. And if you do that um you can just follow that that equation

um to get the number. But but in this problem basically what what what you should do is first let me erase these

erase. Okay. So first you count how many

erase. Okay. So first you count how many what is I interior.

So you have to you know count how many

dots are inside guys. 1 2 3 4 5 6 7 8 9

10 11 12 13 14 16 17 18 19 >> 20 21 22

Oh. Oh, you're not

Oh. Oh, you're not >> um when this happens.

So I think it's 22. So you basically count the interior points. How many

interior points are there?

>> And then you count the boundary >> teacher generation.

>> You count how many points are on the boundary.

>> So can you count the boundary points?

So I have one, two, three, four.

Um, >> is that one in the middle? Count

boundary point.

>> Huh? Which one?

>> Like the >> I think this so tall you're blocking the whole screen.

>> This one. Wait, this one.

>> Sorry. Which one?

>> This This one. Okay. Oh yeah, that that one actually you have to yeah it has to be drawn nicely so you can see if it's so actually there is so

this is 5 by five this one is perfectly in the middle that this one does count and the other two doesn't count so we have one

two three four five so we have five um and interior Yeah, this one you have to count it

carefully. So, let's say

carefully. So, let's say >> the ones lining the hole also counts.

>> Huh?

>> Maybe I count it around. Uh, can you count how many interior there?

>> Let me erase it.

>> Count, right?

>> H >> the ones around the hole. Count

>> around the hole. Yeah, you have to count that.

>> There's 21 um interior um Let's see.

>> So, I have one 2 3 4 5 6 7 8 9 12 13 14 15 16.

Yeah. 21.

Okay.

So, let's see. I +

B / 2 - 1 is + 2.5

and that's 20 2 - 5, right?

But that's including this area.

Okay. So what we did is including this interior area and therefore it needs to be minus one. So that's

21.5.

Okay.

So if you know pix method pix theorem then you can use the this forometer is I plus half I have I have B minus one and

then you can calculate it. Okay. or you

could use the complimentary area calculation either way. Um, and this one is it's just the picture is a little bit distorted. Um, if if it's not then

distorted. Um, if if it's not then easier for you to count.

Okay, that's that. And this one

that's that. And this one just counted.

>> Yeah, that's like a formula.

>> It's like 6 to 12. I don't have the 24 right?

>> It's 24.

>> What happened?

>> How do you count it?

>> Because like there's a pattern. So it's

like >> you need one on each and then you need like two >> three four.

You can count on your own like there's a faster way. 12 13 14 15 16

faster way. 12 13 14 15 16 >> 17 >> You're looking at three behinds two.

>> Yeah. No, like

>> you can just you can just look at like the pattern.

>> No, it is 24 6 7 8 9 10.

>> What if we down?

>> Yeah, guys. Let me No, let me let me explain cuz like there's a pattern >> there's a pattern. So like

>> four it pattern. So like

>> here and then we have one on each side to and we so we multiply one by six.

That's six around here. And now then now now then now then now then now then now then now and we need two on each side like this. 2 * 6 is 12.

like this. 2 * 6 is 12.

>> Three on each side. 3 * 6 is 18.

>> Okay. Now I can solve it. And now

>> so some pro some people already found the ways to do it. So

>> uh sorry who's who's trying to speak online?

>> It's so and and again we have multiple ways of doing it. Okay. So um one way is to say just like Brian pointed out. So

looking at the pattern. So the interior >> wouldn't go see.

>> Yeah.

>> Okay.

>> So this one >> is basically one multiply by six.

>> And then the black one you can think of it as this is two >> and this is two and this is two. This is

two.

So we have six twos. That make sense?

>> So it's >> Oh, I don't know.

>> 2 * 6, >> right? Or 6 * 2.

>> right? Or 6 * 2.

And then the white one, you can think of it as three three.

>> So that's that's how you observe the pattern is 3 * 6. That make sense? And

therefore the black one outside should be four * 64.

>> Yeah, >> that that's one way of doing it. Okay.

>> And another way does that make sense, Chloe? Does that

make sense to you? Okay.

Another way is we can use the uh the principle of inclusion exclusion. Okay.

What do I mean by that?

>> Just count this. So

basically or you could just count it but but I want a more systematic approach. Okay.

>> So because we want we basically want the unions right.

Um so I have six of them and then each of each two of them are overlapping at at the vertex. So

basically on the for the the outer layer so each side there are five. So five * 6 we have six sides but then each of the

the vertex is repeated twice. So we need to subtract that. So we subtract 1* 6.

Okay that's how we get 30 minus 6 is 24.

That make sense?

Okay.

So that's applying the the concept of uh you know principle of inclusion inclusion.

Any questions online people straightforward.

>> All right let's move on to >> next >> I guess >> 1. Oh yeah I don't know. Okay,

>> 1. Oh yeah I don't know. Okay,

>> let's do a quick easy guess.

>> What if you guys didn't sit down?

>> Easy guess. Quick easy guess.

>> E.

>> It's D. No, wait. It's not D. A.

>> Oh, I know these. These are bucket problems. >> I don't know these. I don't know. I know

these. I'm going to solve them.

>> It's a >> Wait. Wait, please call me.

>> Wait. Wait, please call me.

>> Please. Yeah, do it.

>> Yeah, man. We solve this. Is it C?

>> See, I knew it was >> like >> I don't know.

>> Wait, calculator.

So sad.

>> Are calculators allowed on the SAT?

>> No.

Wait, what is the use of calculators?

>> Why would calculators be allowed? SAT

exactly >> is chatting allow.

>> Yeah, sure.

>> Yeah. Like we have >> we're just waiting.

>> But isn't it what is it?

>> It's C. It's C.

>> Wait, what's on the SAT?

>> SAT.

>> That's fair. Yeah, there's probably algebra too >> and some geometry too.

>> I said it, >> but not not this difficult >> you spend trying to figure out what each button on a calculator.

>> I mean, if you can do these problems, SAT will be very easy for you.

>> Except the reading part >> seen a lot of >> I think SAT problem is that you have to understand learn the vocabulary words for the SAT

like you have my art teacher. It's so

art. I don't know.

He's not even Japanese, but his whole thing is about his whole room.

>> I've seen I've seen a few of like the SAT math questions. They're not that bad.

>> Mhm.

>> Me too.

>> Daddy's now >> to you.

The beef is not >> Yeah. This

>> Yeah. This say I'm slow.

I think the fact that they're all 1,120 ml bitters is not really that necessary.

>> Yeah.

>> Not need to know that much.

>> Let's see. Wait, what?

>> Giving you extra information.

>> Yeah. Yeah. So, I have to >> observe and you know make a smart decision. Sorry.

decision. Sorry.

>> I need reciprocal.

>> I think I just stepped on the device.

>> Okay. Online people has got some answers.

>> See?

>> Okay.

>> Some people got C. Yeah.

>> All right. Let's take a look at this problem together. Okay.

problem together. Okay.

>> So, well, >> the uh the most straightforward.

>> Okay. Maybe maybe we'll have some online people explain it. Uh Jolene, can you explain your method?

>> Jolene.

>> Um well the in the first picture there is 2 over7 * 1,120 milliliters of orange juice. So that's

um 320.

>> Okay. So you did one. Okay. You did one times 7 2 over 7, right?

Is that what you did?

>> Yes.

>> Is three 300 what?

>> 20 >> 20. Okay. And then

>> 20. Okay. And then

>> in the second one, it's 1,120 * 3 over8.

>> Mhm.

>> Which is 420.

>> 420. I trust you.

>> And then you add these together, which is 740.

>> Okay.

divided by >> divided by um 2,240.

>> Okay. And that's how you get >> 3712.

>> Yeah.

>> Okay.

>> Yeah. So this method is correct but it involves a lot of calculations and it therefore it can be >> you know add them.

>> Yeah.

>> Then multiply by one.

>> Okay. No doubt.

>> So, so this is correct, Mat, but it's not probably not the best. So, let's uh see what Brian >> So, you can like find the common denominator of the two books.

>> Yeah, >> because they're all just proportions.

>> So, basically Brian's saying we can find the common denominator of these two numbers.

So, seven and eight, right? So it would be >> 56 or multiples of 56 and then >> so it's like >> and that ruled out basically

three answers right does that make sense to online people or or to people here like why Brian is looking for common

multiples of seven eight and so we we have two left now how do we choose >> well first well I mean like First we have to like get the fractions. So

>> Mhm.

>> two 2 over 7 with the denominator 56 you multiply the top by 8 which is 16.

>> Yeah.

>> Then from 3 over 8 you multiply the top by 7 which is 21.

>> Okay.

>> Oh yeah I got that algebra.

>> That's cool.

>> 21.

>> And we can add these two together >> to get 37 over 56. But we have to >> divide it by two.

>> Yeah. So add them up and divide by two.

And that's how you get 37.

>> Okay, that's very good. So basically

what what Brian is suggesting is that because the these two are the same volume. So it doesn't really matter

volume. So it doesn't really matter whether it's 1120. It can be any volume.

And the first one is 27th and the second one is 38.

Right? And now when they are added up basically the total volume is doubled. So

therefore you need to divide this by two. That make sense?

two. That make sense?

So this is the proportion of the substance given one volume. And this 3/8 is the sub substance in another given another unit volume. But now when

they're mixed the unit volume is doubled. The total volume is two.

doubled. The total volume is two.

There's two unit volumes. And therefore

the total answer should be this. And

therefore it should have a common denominator. You know the denominator

denominator. You know the denominator should be a multiple of seven and eight.

Right? So it it immediately we have these two results. And you if you do the calculation you get this. Now we can even make the choice without doing the

calculation. You know why?

calculation. You know why?

Because we know that when we mix these two things, right? I mean they have the same volume. And when we mix them, we're

same volume. And when we mix them, we're basically taking the average, right? And

that means the to the final number, the final fraction should be in between these given two numbers. It cannot be smaller than the smaller of them and it

cannot be bigger than the bigger of them. Right? Does that make sense? When

them. Right? Does that make sense? When

we mix these two at the end, we get a a kind of average concentration.

So the average concentration should be in between 27th and 38 and according to that we can immediately

root out D because D is too small. It's

it's much smaller than you know 27th cuz 27th is like 16 over 57 56

right. So 5 over 56 is far less than 15.

right. So 5 over 56 is far less than 15.

Oh sorry 16 over 57 which is 27th and therefore this can be ruled out as well.

So we're left with only C is a valid choice.

>> What a waste of some good orange juice.

Okay. So if you use this method, you can quickly nail down the answer in in you know 30 seconds.

Okay. So so we're trying to you know inspire you to think more about you know can you come up with some easier methods to ensure that your answer is correct.

And there are two purposes. One is you know first it helps to know multiple ways to solve some same problem. And

second, if you can identify the fastest route that will save you time, which is precious in this exam and you can use the rest of time to deal with the more difficult problem or you can have, you

know, use a different way to double check this answer. Make sure this is correct.

Okay.

>> So the answer is >> the answer is C.

>> Yeah.

>> Any questions, comments from online people?

>> Almost all C. Oh, it's always No, not necessarily.

>> Same. No, I like >> Okay, >> let's take a look at >> who sees the entire thing. I thought I

>> Okay, they use the CVS method, which is basically what uh Jolie, >> right?

>> All right, let's take a look at this.

into five.

>> Hey, just type it there cuz it's really noisy here.

>> Type it in the chat if you got the answer.

>> Like stars and bars.

>> Yeah.

Hey, I don't >> you and >> double check your answer if you >> you're a Brawl Star >> if you finish it or think about is do you have any other ways to solve the same problem?

>> It's a combination.

>> Okay, technically we're pretty much just distributing two principles.

>> Wait, it's C five.

>> Technically, we're distributing two.

Huh? What are you >> solus?

I like stars and bars.

>> How do you do it?

>> I have I hate stars and bars. I can't

give >> I didn't really like you in stores and bars. I just did like oh

bars. I just did like oh >> okay yeah we can do it without stars and bar or we can do it I mean I think if you understand stars and bar is pretty

straightforward and it's less >> less brain consuming >> my teacher just do How else?

Um, well, the first part, Alex, the first part is correct. Then how do you get I mean, so first you did 12 - 5 * 2

is 2. And then how do you how do you get

is 2. And then how do you how do you get 2 * 5?

All right. online people. Shall we look at it this together or >> no one got 10?

>> Oh, some people got 10. Yeah,

>> like way too small.

>> Addex also got 10.

>> Okay, >> we have to claim that because 16 is too big.

>> What does that mean? Wait, what is that?

>> It's a day.

>> No, I don't.

>> 50.

>> Oh, I I do 50.

>> The second digit is >> Oh, right.

>> Yeah.

>> What's the second digit?

>> Wait, so you don't want the people to >> Well, I mean, they're still thinking.

They're still some people are still thinking. So

thinking. So >> they are there are smart.

>> Okay. Yeah, Victor got it. Um,

>> of course got it. Okay,

>> I got it faster than Victor.

>> Good. All right. So, in the interest time, let's look at this problem together. Okay. So, I think most people

together. Okay. So, I think most people get it that, you know, we can first predistribute the two pencils, right?

So, everyone gets two. So, that means we have, you know, two left. So now is how do we distribute two pencils to five

people and each like you know some people could get zero right so that's first part every everybody agrees so it's you know

12 minus that 2 * 5 is two okay now to distribute two pencils to five people some people can get zero how many ways how many different cases different scenarios

>> so it could be two and the others being zero right?

>> Yeah.

>> So, basically, you just pick one out of five to get two, then you're done.

>> It's there's like 10 options, >> right? So, that's five options.

>> right? So, that's five options.

>> That's five options.

>> Now, another way is one person get one, one person get one, and the rack gets through zero.

>> Right? So, two cases >> case work now. Oh, um, you're not seeing anything.

It shows that >> the whiteboard is really living up to its name synchronized with the other white.

>> So okay so the second case is one person get one one person get one

and the other zero.

So the first case is two and the rest is zero. So that's C51 is five. Okay. And

zero. So that's C51 is five. Okay. And

second is basically we pick two out of five people each person receive one. So

how many? So it's C52, right? So it pick two out of five to

right? So it pick two out of five to receive one each and that's 5 * 4ide by two and that's 10. So it's 10 + 5. So

that's C. Okay. So this is doing it without stars and bars.

No, you can also do it with stars and bars. Everybody understand this? Any

bars. Everybody understand this? Any

questions?

Yeah. Yeah. Jordan, you're you're okay. So another way is stars and bars.

okay. So another way is stars and bars.

So basically we're saying originally so so alo we distribute we predistribute to to each person. So so now we have just two pencils to distribute. So this

is basically we have two or previously we said x1 + x2 plus x3 x4 x5 right is

equal to two and then each of x i is greater than or equal to zero.

So meaning some of them could receive zero. So now this is standard stars and

zero. So now this is standard stars and bars right. So this is basically n= 5

bars right. So this is basically n= 5 and r equals 2. And for stars and bars,

the equation is m + r -1 and then to the uh n minus one, right?

Or or to the Yeah. So this is c. So 5 + 2 - 1 is 6 and then n minus one is is 4, right? Or you can you can just use r.

right? Or you can you can just use r.

It's the same.

So, so this is 6 * 5 / 2 is 15.

Okay. So, if you know stars and bars, it's just a matter of plugging in the equation.

Does that make sense?

>> Yeah. Yeah. But, but but I I hope you understand this method.

I care more about or you can you can do n + r minus one to the r. It's the same.

But I think this is easier to remember because it's basically like you have you you partition into five parts and you need four bars. So select four bars out

of this many.

Okay. Uh online people does that make sense to you?

Assuming some like you have learned stars and bars. If not, you know, we also have previous lectures. I I can, you know, >> can I finish the bathroom?

>> Yeah, sure. Yeah, we can take a short break and bathroom break and then we'll continue. Okay.

continue. Okay.

All righty. Any questions online people?

Joden, does that make sense to you?

Okay, good. You've learned stars and bars.

Okay. and and Jayen as well.

>> Jayen, what which uh which grade are you in?

>> Uh sixth grade.

>> Oh, great.

>> And Jodine is uh >> math.

>> Yeah, she's in eighth grade.

>> Okay.

>> All right. Yeah, let's take a couple minutes break and then >> continue. And then the next problem

>> continue. And then the next problem we're going to deal with is this. And if

you want to take a look, you know, feel free.

I think Ellen.

>> So Ellen hits N eight and 10. Wait, do

we just name this if we want?

>> Okay, it's not Ellen. Ellen does.

>> Yeah, it's not.

>> How about let's how about Ben? It cannot

be banned because all like uh one number >> does one.

>> How about six?

>> Oh, >> I think it's because I think it's Alice know five and seven and then >> you know

>> and then does two and six and then Dave does four and fits perfectly. So

>> yeah, >> wait and multiple people get the same number.

>> No, like do you know a >> what do you mean? They never said that hits the target region target region with a different value.

>> So okay so when this angle is still here >> oh This is a straight dog.

>> I don't think that's how >> angle A will be 30°.

>> Yeah.

>> And then it's 30°.

>> You know, you know, in rectangle, >> then this line So the bonds,

>> okay, that means >> they're not good. They're just bad.

>> Math during our break.

>> And then this question is >> like then this would be 30.

>> This is 30. What is it?

30. What is it?

I don't know.

>> Okay. Shall we shall we continue?

>> Um, okay. Sure.

>> Because she gets too.

>> And guys, please please focus. Okay.

>> So, I hope that you you think about the problem and >> think about the problem at least that way. That way when when I explain it,

way. That way when when I explain it, you could benefit more because it doesn't help if if I just tell you the solutions, but you have to think about it yourself.

>> So So each of these problems I I think of each of these problem as a as a kind of a piece as a piece of art, right? So

mathematic is like the the gymnastics of the brain. Okay, mathematics like

the brain. Okay, mathematics like gymnastic brain and these are Is is the answer Cindy?

>> Um, >> okay. Even

>> okay. Even >> Yeah, double check your answer. Yeah, I

mean, >> see if it's consistent.

gets to seven and seven playground when your name problem >> I'm so

it's bro >> it is >> it is so >> hey folks folks that's that's focus Okay,

>> let's focus, >> Alice.

>> All right, Alice, you're the answer to the math problem apparently.

>> Yay.

>> I don't know.

>> You don't know.

>> I told her a statement and she says she doesn't know.

>> You think you guys realize math problem?

>> Yeah.

>> Oh, really?

>> Do that.

>> That's a good observation.

celebrity is my >> to simplify like the student's life so that your name is

not at all Brandon and how confident he is in his wrong answers >> and we don't have the time to check him so we just put it there and it's wrong but what you get for the first three

answers we got C >> the first answer was the first uh answer >> I just remember this problem with >> I just remember that only one

>> there's like a five by >> no it's like 20k Let's focus on >> let's focus on this problem. Okay folks

>> focus on this problem >> folks let's focus on this problem. We

can discuss these afterwards. Okay.

>> You ask a problem. Can you solve it?

>> Yeah.

>> From yesterday. Okay.

>> What is one plus one?

>> Wait.

>> Wait. Let's let's do that for that.

Yeah.

>> The problem. You don't remember.

>> I don't even think you even had that problem.

>> My computer has all the problems. You're a problem.

>> No, I'm asking.

>> We're all problems. >> We all have >> started talking about George was the one who sold us yesterday. I remember a guy that was

yesterday. I remember a guy that was like >> short in our group >> and he kept talking about kids at his school autism.

>> Autism like me >> and he kept talking about Down syndrome and >> we don't know each other before this.

>> I love him.

>> I know him too.

>> I know your I know his sister.

>> I knew you know.

>> Oh yeah.

>> What do you mean? Oh yeah. Not his

sister. Is it your Cuz I'm like >> My sister's in West Point.

>> Hi guys.

>> My sister knows you.

>> Yeah. Ninth grade.

>> He's on.

>> Yeah. Yeah. Right.

>> I remember it was like >> to 450 I think.

>> No, that's like 40 minutes.

>> Yeah, say 20 minutes.

>> Oh, I live like >> 35 minutes away.

>> Live in New York.

>> Oh, I live 10 minutes.

>> I live too.

>> Oh, yeah. I do not remember >> knowing how to do this like fail or old.

It does take a long >> wait. Have you guys solved the problem

>> wait. Have you guys solved the problem parting out?

>> I don't think we >> No.

>> Okay. That means you share it everything. But what's the what's the

everything. But what's the what's the least common multiple of four, five, six, and seven?

>> Six, seven.

>> How does Victor know about a place that's 2,000 miles away?

>> I don't know. How did he just like?

No.

signific take a look at this problem together.

Let me see. I I think there are some Oh, Victor got a Oh, okay. It's Alice.

okay. It's Alice.

>> So, Michael got C and Jolene got A.

Okay.

>> He lives in >> Right. So,

>> Right. So, >> okay. Yeah, let's let's take a look at

>> okay. Yeah, let's let's take a look at this problem together. Okay. So,

>> yo, >> oh, Victor, would you like to explain your way?

>> Well, so first I found out that Ben has to be three and one.

>> Okay, very good. So, Victor made the observation that so basically out of these Ben is the smallest, right? So the

m number of possibilities of addition is is probably the the smallest. So in fact four can only be 1 + three because each number has to be different,

>> right? So that's the only option.

>> right? So that's the only option.

>> Next.

>> So then um and Ellen has to be 10 + 8 >> and Ellen the largest also only has one option. So that ruled out a lot a lot of

option. So that ruled out a lot a lot of other possibilities.

And then >> so then um Cindy used up that all of that. Then Cindy has to be 6 + 2 because

that. Then Cindy has to be 6 + 2 because it can't be 1 + 7 five um six or and it can't be 4 + 4. So Cindy is 6

+ 2.

>> Yeah, because we already used one and three and it cannot be four plus four.

So the only option is 2 plus 6. Very

good. Also and and also Andy Atlas has to be 7 + 5 because there's no other way to make 12.

>> Yeah. So right now after all these numbers are taken, right? There are not not many ways to make 12 anymore. So the

only option left is 5 + 7. And that's

how Victor found that is Alice who has the seven.

>> That's hard words.

>> Huh?

>> Dark.

Delta is not not like that.

>> This is more of like a target like the a red and white target.

>> Oh yeah. Well, I mean it's a math problem. They made up things. They made

problem. They made up things. They made

up a lot of things >> like monster eating people, you know, like I mean it's math. It's

>> No. Why is the monster give you exactly three strikes to visit?

But it could it could be a it could it could be a game with monsters, right?

>> And then the condition is that if you hit the all these three >> points, then the monster will eat you.

>> Starting zero and I need to get at 10.

And then like in the middle nonviolently says, but there's a monster at 63 that'll eat you and burn you alive if you cross.

>> No, I want this like manifest. Well,

like I said, it could be it could be it could be a game, you know, like an electric game where >> again it's made all made up. Okay, let's

let's look at this next madeup problem.

>> Larger than two.

>> Really?

>> Thank you. There's no answer that >> I remember. But then again, >> yes, cuz I know that 1.17 is leaves a

remainder of three when divided by four, five, six, and 10.

>> Okay. What's

>> What's the LCM of four, five, six, seven?

>> We just multiply them or like it's like two.

>> Okay, remember the topic of this class is how to solve AMCA problems in seconds.

Well, I I would say solving problem >> it's B. It's literally B.

>> Mhm.

>> Because we you have to find the LCM of all those and then you get three.

>> No. Yeah. No, you all.

>> All right. So, we have people who already found it and >> are you getting >> B? Okay. How do you get it?

>> B? Okay. How do you get it?

>> B. What?

>> Yeah. How do you get it?

>> Fine. 4 4 * 5 + * 6 * 7 is 420 >> I think. So

>> no it's not >> 32 * 5.

>> Okay 210.

>> So basically you need to find the least common multiples right of these and then plus three.

>> So the least common multiples of these would be four * 5 * instead of six we do three because 12 is already a multiple

of six.

>> So that gives you 420.

>> Yeah but you have to add three is 423.

So it's B. Okay.

>> You are >> so tough.

>> We are boys are >> so this is in seconds for >> and next one.

>> Why would they have to write four nights in third form?

>> Why would they do >> Oh, I hate that someone >> Why do they have to make the problem look as convoluted as possible?

>> We are seated in the seven points. Why

are they stand?

>> Okay.

>> I mean, it's it's fairly easy.

>> Yeah, >> it is fairly easy.

>> There's empty chairs. Sit down.

>> Oh, quick educated guess. Quick educated

guess. It's B.

>> Okay. It's not B, bro.

>> It's D.

>> It's E.

>> Is it D?

>> If there are six empty chairs.

>> Oh, yeah, it is D.

>> Is it?

>> Yeah, it's four.

>> Okay, so five.

It's just me or the BCA problems on like this week's homework. Look at that alpine kid at the beginning.

>> This demorous kid, bro.

>> Aren't you? Oh, you're Harper.

>> Hogworth is so tough.

>> What does Harper fat?

>> Hogworth has nothing.

>> Oh, we can't wait to see Alpine. We have

a single restaurant and a single.

>> Well, at least Alpine has mansions.

>> Okay. We have

people got it already.

They have like >> Okay, Jolene, how do you calculate that?

>> Stop talking and focus on the problem.

>> Yeah, let's focus on the problem. Okay,

>> it's B. No, it's C.

>> Okay, Jolene, how do you calculate it?

>> The problem. Um well 4 11th of the chairs are equal 16. So then there's 44 chairs in total. So then the number of

seated people would be 7 over 11 * 44 which is 28. And then the 28 people are 4 9ths of the people in the room. So

then there's um 63 people.

>> Okay.

>> So I think you did exactly this right.

So >> um >> so you first get 44 chairs in total >> and then you know the number of seated

people are 28 and then and then you use four four 9ths and yeah that's good but um remember this is the topic is how to solve it in seconds right

>> so >> is there any quicker way >> I Okay, take an educated guess

and 16. So 11

and 16. So 11 is 4. So 4 *

is 4. So 4 * >> 4 * 11 is 44.

>> Well, 4 * 7 is 28 and then 28 / 4 is 7 and then 7 * 9 is 16. Okay, that's still pretty that's pretty more a lot of calculation. Okay, my way of doing it

calculation. Okay, my way of doing it >> see n I see.

>> Okay, so it's se it's four ns of the people that means the people number of people is a multiple of nine, >> right? Multiple of nine.

>> right? Multiple of nine.

>> So there is only one multiple of nine which is D.

>> Okay. Okay. I got a question. What if

they chop people in half? Okay.

>> But you have to chop people in nines, right? Nine portions.

right? Nine portions.

>> Still while I can put this stick so I can >> exactly four months.

>> Okay. So So again, you know, you have to make good observations. Okay. And and

and find out what are the key things. So

in this in this problem, I think nines is the key. And that's how I found the answers in seconds. And you can do it too if you pay attention. Okay.

>> But I still don't think it'sable because in one of my RS questions like they have things like there's one point.

>> I mean since you can get this in in seconds you can double check. You have

plenty of time. You can calculate oh 9 4 9 of 63 is you know you can calculate that. Okay.

that. Okay.

>> You have multiple time multiple plenty of time to do the variations.

>> Okay let's move on to the next.

Oh this >> shaded area is nonexist.

>> Yeah, there's no shaded area. So, it's

undetermined seconds.

>> Done. E.

>> Question 24. How is it the hardest?

>> Yeah. Okay. So, the shaded area is this.

Okay.

>> Okay.

>> Um, two squares next to each other.

>> Yeah.

>> The largest assume that the the okay find the shaded area.

>> Oh god.

>> What is the shaded area?

>> It's right there.

>> Big triang.

>> I wonder why >> wait2.

Why can't I be >> 20 and a half?

>> Yeah tournament.

>> Wow. If there's not enough information, then you can say >> we don't.

>> But there is the area of the small.

>> Huh?

>> What is the What is the square root of 5,000?

>> What?

>> I don't think that's my >> 5,000.

>> Wait, no. I'm using

I love having to solve forot.

>> I love doing 37. Am I right, guys?

boxes like subtract something 150 200 300 and divide I think I have enough >> Alex. I don't see the answer choice of

>> Alex. I don't see the answer choice of zero.

>> [ __ ] >> Wait, zero.

>> Able to tell the answer by >> the product.

>> No, I think >> sending all your blood to your brain, you know. So,

you know. So, >> I spelt little.

>> Let me just double check.

>> Yeah.

>> Oh, I see. Okay.

>> Yeah. I I literally didn't put it so people >> Let me see.

>> Guys, I'm What's your question? I see

the >> true enlightenment. We must first achieve our brightest >> Oh, I thought it was the blood there.

I feel a little lightheaded here.

>> You're going to get dizzy pretty soon.

>> No, really.

>> Why don't you just get Why don't you just record it and then flip the phone over to sleep?

>> That's a TV screen. How are we going to flip the TV screen?

>> TV screen.

>> Take the picture and then flip the phone.

>> I never realized that.

You have a computer. It has a camera on it.

>> Draw it on your paper and flip your paper.

>> You don't need a phone. You don't need to.

>> Exactly. You can try. Yes.

>> Camera.

>> It is not the same.

>> All right. Shall we uh look at it together?

>> Let's go.

Um, yeah, that's a good guess. So, okay.

So, we already know the side of the larger square is 20, right? So, this is 20 and this is 20. And we know that total area of the square is

>> 20. No,

>> 20. No, >> 20 times 20, right?

>> Yes. A square

>> 20.

>> That's That's this.

>> Oh, bro. Oh yeah,

>> this happens every time.

>> Sorry, >> cuz the first the first one pops up is that >> synchroniz.

>> So 20 * 20 is this area. Okay. Now 1

half of it is 200, right? And that's one of it turns out to

right? And that's one of it turns out to be one of the answers. Now is the question is can we justify like one half of it or one half of it is equal to the

shaded area. Are you looking?

shaded area. Are you looking?

>> So one half of it >> just trust me >> and one half of it. Now this half this half actually has something in common

>> with the shaded area >> are you listening? So this half of triangle actually has something to do with I mean there something in common

with the shaded area. Now we only need to compare the difference. Does that

make sense?

>> So basically this area versus this area.

If they're the same then it's 200. Make

sense?

>> Imagine how this part can go here this and go here.

>> Well but they're not identical. They're

not congruent. Wait, wait, wait. Isn't

this box like isn't the length of this box half of this?

>> No.

>> Wait. We don't know that.

>> We don't know that.

>> Okay.

>> Now, >> Alex, you're right. But, you know, like why? We We're trying to justify it.

why? We We're trying to justify it.

>> Okay. So let me let me uh but but you you guys have have learned about um

like the area of a triangle is just the base times the >> the height, right? So if we compare these two triangles, what what what do

you see?

So let me rewrite it.

Okay, Charlene. So, if we compare

Charlene. So, if we compare this one because this one we know is 200 versus this one.

What do they share?

>> They share that little part.

>> They share this this side, right? And so

if you rotate it, so this is the base.

If this being the base, what's the height? What's the height

height? What's the height >> of >> the height of this one versus the height of that one? Are they equal? Are they

equal?

>> They're equal.

>> Both.

>> So it's same >> if you can prove as they're equal. Make

sense?

>> Now how do we prove they're equal?

>> So we can connect the smaller square.

That's why the smaller square exists.

Okay, so we connect the diagonal vertices of the the smaller square and we know that this is 45 degrees, right?

And for the larger square, this is also 45 squares uh 45°.

And therefore, because these two angles are the same and therefore these two lines are congruent, >> these two lines are parallel to each

other right?

fluid lines >> and and therefore the the distance of these two >> dots, these two vertices on the smaller

square to the large diagonal of the larger square >> is the same. The distance is same because these two lines are parallel and the distance is basically the height.

>> Does that make sense?

>> And also quick question. So we just showed that these two triangles they have the same height and they have the same base and therefore they have same area.

So what we need to look for the area is exactly half of the larger square. So

it's 200 is 200.

Okay.

>> Yeah.

>> Why do >> the Play-Doh?

>> I almost touched.

>> Okay. Does that make sense to online people?

>> Why would >> Are there any questions?

>> Alex, does that make sense now?

>> Teenagers.

>> Okay, maybe you just eyeballed it. But

oh, I hope that you can find a um now you understand how to prove it rigorously.

>> Do you know you're not g like the desk?

>> I have no questions. We're going to move on to the next one. Okay.

All right, guys.

>> More geometry problems. >> Me, too.

>> I like it. I just need to look at it.

>> So, in a way that makes blood rush.

>> What is this?

>> In a way that makes you lose circulation.

>> This doesn't help though because there's no like or anything.

>> Yeah, it's just going to be >> There is that. Two of them are 20 by 20.

>> Oh, sorry.

>> It's a D.

>> All right. Okay. I'm going to find out how >> Oh, wait. Wait. I see.

>> This is not It's not going to help us now.

>> Oh, I see. Wait. If this is >> this is kind of upside down. Ethan, it's

the same picture.

>> When you do that, do the neurons like rearrange themselves so that you can see the image better or something?

>> Try this.

>> Has to be 10. That means it's >> the huh become >> so usually it helps to have these dotted lines.

>> Very well. I see the rear end of something.

>> Got to find out how it is.

>> Oh, wait.

>> Oh, wait. Wait. 100.

>> Yeah. Wait. It's pretty obvious because you just know that like, you know, these are 20 by 20. So each one is 400, but they intersect right here, which makes the lose 100, so it's 50.

>> Easy.

>> You just need to turn in a different direction. Literally life.

direction. Literally life.

>> I did it the easiest way.

the other focus >> shing let's focus on this this is one of the homework problems >> the way I eas immediately the answer >> so these are 20 by 20

>> the 20 by 20 right >> people so okay uh they're each 400 but we intersect at this little area which

is exactly 100 they lose 100 area so 800 - 100 is 700. Okay, I agree with whatever he says for you too.

>> I just for yourself.

>> How was that deja vu for me?

>> I don't know. I didn't know what deja vu is.

>> Deja vu is first national deja vu is >> no school teachab.

But they don't teach us what deja vu is.

No me too.

>> Yeah.

>> I'm taking algebra 2.

>> I'm taking algebra one.

>> Well, you guys are No, my school we do algebra, algebra 1, geometry.

test the core. Daniel

>> one just like normal calculus and >> the rare ant thing again.

>> What >> in school?

>> Wait, >> what rare?

>> Middle school. I'm not learning.

calculus your school is not teaching you calculus that's something else >> sorry I didn't understand the question Alex >> question

>> all right folks that's >> focus on the problem okay we only have like 15 minutes left >> is it >> I I honestly don't know question

>> you don't you know I just don't know what to do.

>> You don't like >> So, okay.

>> The key is in the first number.

>> Yeah.

>> Oh, the di >> the first number will make your life a lot easier because >> circumference divided by >> All right. So, online people, shall we look at it again?

>> Together. Okay. So, I think everyone agree that the shaded area is the total area minus the circle, right?

That's >> okay. So, so the shady area is the total

>> okay. So, so the shady area is the total area, right? The total area t minus

area, right? The total area t minus p<unk> r², right? So, it would take the form of something minus, you know, pi *

an integer, right? Because they are all in this form.

>> Now, the total area, what is the total area? Well, this is very I mean we can

area? Well, this is very I mean we can apply the principle of inclusion exclusion, right? So if we draw out

exclusion, right? So if we draw out these dotted lines the small square here. So the total error is just

square here. So the total error is just this square the bigger square. So 20 * 20 and plus the other 20 * 20

subtract what it has been double counted. Right? So 10 * 10 is doubly

counted. Right? So 10 * 10 is doubly counted.

>> Does that make sense?

>> Wait, you're not saying it?

>> Why don't like >> No, I think it's cuz there's like two different They're unsynchronized.

>> There's two sets of Yeah. Well,

>> there are two sets of notations and You guys Shanghai Shanghai or Shanghai?

>> At least you got to listen to it again.

>> Shanghai like American.

>> So total area minus p<unk> r².

>> Okay, I'm rewriting it. Hi guys.

The total area is shine. So the total area minus the the this this circle, right? Pi r square. So they all take

right? Pi r square. So they all take this form. And now the total area is

this form. And now the total area is just a union remember. So we apply the concept of inclusion exclusion. So just

the ones of the square plus the other square minus the intercept the intersection. So plus 20 * 20 and minus

intersection. So plus 20 * 20 and minus the the intersection is 10 * 10 -<unk> r².

Right? So the first part we already have 400 + 400 - 100 that's 800 700 minus p<unk> r square. So, we don't even care about what pi r square is because

there's only one answer that has 700 in it, right? I mean, you can spend time to

right? I mean, you can spend time to figure out r square is 50, but that's not necessary because you know 700 already gave us the right answer.

>> Everything like >> uh Jolene said, don't the two squares also intersect? Yes, that's exactly why

also intersect? Yes, that's exactly why we need to subtract the intersection.

which is 10 by 10.

Does that make sense?

And and there's only 700. There's no 600 or you know 650. There's nothing close.

Are there any questions from online people or from here? Does

that make sense now? So that's how you do it fast is you know we we only need to calculate 400 plus 400 minus 100 is 700. Then you get the answer. That's how

700. Then you get the answer. That's how

you did it in seconds. Okay.

And yeah of course you can spend time to figure out what what the you know the D is and etc. But that's not necessary.

All right. This one.

Oh, I love these.

>> Oh, it's an isosles for sure.

>> Yeah, we kind of >> Yeah, let's focus. Yeah.

>> No.

Area.

>> Yeah. The area.

>> So, the base is 42.

>> Mhm.

>> The height is somewhat around 20.

>> Mhm.

>> 21.

>> Is 21 something 29 like a Pythagorean thing?

>> That's a question. It's a good question for you.

>> Time to figure out figure that out.

>> Figure out quickly.

I know no better quicker way than to just multiply them.

>> There is a quicker way.

>> So it's definitely not 100.

>> I don't think it's 500.

>> It's bigger than 500.

>> There's no way. There's no way the height is >> answer. So

>> answer. So >> it's definitely not 4.

>> Not a single. Yo, I'm I'm saying >> there's one single two that appears in the answer choice.

>> 20. Never mind. Well, how? There's no

way the height is 20.

>> Yeah, the height is you just did divided by two.

>> Oh, yeah. Yeah.

>> Oh, yeah. It is.

>> Y has B.

>> Mhm.

>> Please remain Christian.

>> Okay. Victor got B as well. Okay. So, so

yes, you can do Pythagoran theorem. So,

okay, let's do it together. Okay,

>> just >> so this is 29 >> and this is 21, right? And so the height

is basically square<unk> of 29^ squar - 21.

And how would you do that? Well, it's a square - b square. So, it's a + b * a minus b, right?

So, it's 50 * 8, which is 400. So,

square<unk> of 400 is 20.

Okay. And then you use half of 42 * 20.

So, basically 21 * 20 is 420.

Okay, make sense.

And so a square minus b square is very powerful. And another way is even though

powerful. And another way is even though we cannot do this and let's say you know in the future it's not so easy to calculate through the pythagoran theorem

then what do you do? So we can use this herren equation. Okay. So, Heron's

herren equation. Okay. So, Heron's

formula basically if so basically we calculate first >> what >> the uh half of the of the parameter

>> for very simple >> so it's very simple um so calculate half of the parameter so 29 + 29 + 42ide

by two is 50 and then you use this so let's say this is s okay smaller s then total area

Um the area is just square root of s * s - a s - b s - c. Okay. So you just

calculate this square root of that and that also gives you 420. Okay.

Um all right.

So that's that's the heron theorem.

Basically s is a + b + c / 2 and then the total area is just this square root.

Okay.

So it helps to remember this barometer in the future.

>> You should have no problem.

Huh?

>> Oh, you want to write down this formulas? Yeah.

formulas? Yeah.

>> It's heron's formula.

Yeah. Square root of that.

>> Okay.

>> All right.

>> This is simple.

>> Okay. This is not that fast.

>> Yeah.

>> Hey guys, >> 12.

>> We're almost done. A what's the area of AMC?

>> Yeah.

>> A >> I think Aals.

>> Oh, you're just realizing >> and then you >> Yeah, this is really easy, right? Just

base times the height.

>> AMC 2.

>> Wait, is it 12? Is it? Yeah, because

it's four and then six.

>> So the answer is 12, right? So it's

basically base >> base times the height divide by two.

Okay.

>> All right.

>> And this one. Okay.

>> So the diameter of each of this half circle is eight. So find out the shaded area.

>> What is that?

>> What? But there's an area outside of the box.

>> No, no, no. I mean, I'm talking about the the, you know, the light.

>> Okay, let's approach this.

>> Wait, actually, that's not that bad.

>> Yeah, it actually isn't that bad. It's a

little trick.

>> It's a little strange what it is that >> Wait, the shaded area is >> displaying not contain shaded area.

>> Isn't the shaded area just a >> It's 14.

>> No, it's this one.

>> For sure if it's 14.

>> It looks like it's 14. Despite the

figure not containing any straight lines, it looks pretty simp.

>> Wait, what's the area of the shaded >> the area?

>> Yeah. What? Like what's the question?

>> The shaded area. Calculate the shaded area.

>> 16 is that the whole area is 64 >> just by eyeballing.

>> It looks like 16.

>> Just looks like >> Yeah, because it looks like it's 14.

>> Looks like Can you prove it?

>> Can you prove it?

>> Since this is the main point of the entire circle.

>> Oh, wait.

>> Shift this down here. It does look happen since this midpoint and move it down here. And since this is the

down here. And since this is the midpoint and this square here is the same.

>> Mhm.

>> We can form this triangle.

>> Can we just like move this here and make a circle?

>> We can we can chop this in half. But

what he's saying is chop this in half.

Move it over here. We have this triangle which is exactly what to make a circle.

>> So I think some people already >> very good. So, so Brian found it. So,

basically if you chop >> I think you found it.

>> I think >> if you chop this in half like >> you have to bring >> and then you you can bring and >> why did that >> rotate it >> here? Rotate it.

>> here? Rotate it.

>> What is the word for that?

>> You see that?

>> What do you mean rotate?

>> Okay.

>> Okay. Rotate this part over here.

>> See, I'm actually and then rotate this part over here. Can you see that?

Um, Alice, does that make sense to you? So,

we can chop the upper part into two and then rotate them to fill this part.

So, it becomes 1/4 of 8 * 8, but that's 16.

>> Make sense?

>> So, good.

>> So, a lot of these geometry problems, if you figure out those dotted lines, I'm >> then it helps.

>> Is this like question 25?

>> I do remember this one because I didn't.

>> Yeah, >> leaked footage of question 26.

>> Remember, >> guys, this is leaked footage of question 26.

>> It's like getting This is like clicking a button, clicking one of those.

>> Wait, it's like What are you talking about?

>> My legs that says preand >> is it?

>> Yeah.

>> One look at it and I know it's >> great. I said it before.

>> great. I said it before.

>> One look at it. I'm telling you.

>> Oh yeah.

>> No, I see Steve. Let's No, I guess I think it's trust me.

>> I I think it's that one.

>> I think it's that one that that I'm pointing at.

>> Okay, you're pointing at number A. So,

you guess A.

>> Number A.

>> Number A is crazy, bro.

>> A is a number. It's the

>> A is number one. A is the first letter of the alphabet.

>> A is number one. Wow.

>> What?

>> It's number one.

>> Dylan, you can leave now.

>> 30. Yeah, it's E. You know, I find that funny because E is the most common letter >> E is not a grade. It should make >> Oh,

>> for some reason >> because everyone thought >> A.

>> No, they took out E from the grades because like everyone thought it was excellent.

>> Wild math.

>> Yeah, there was a awesome.

>> You look at it that way.

Yeah, it's absolutely fantastic.

>> No, >> they took away a plus six.

>> Why? Yeah, my school's coming >> because they think there's Okay. Some people got it B. Yeah.

Okay. Some people got it B. Yeah.

>> All right. Some people got it J like 16.

>> So, >> okay.

>> I was put on your whole class.

>> Oh, wait. You're a drug.

>> 16 was the for the last problem. Okay.

Yeah. Good. All right. So, this problem, let's do it together. Okay. Um,

someone's time. So, what I would do is basically we can, you know, I hope maybe that's also how you did it. Basically,

we can draw these dotted lines to complete the triangle. Bye.

Um so the total area is basically the area of the triangle minus these two and

this is 60° and 60° so it's 180 120° in total out of 360 right and that times pi

r² r is 2 and so we have an area of a triangle minus 1/3 of 4 pi

Right? And just by this even without

Right? And just by this even without calculating the area of the triangle we know that it must have a characteristic like this and that give us the only

choice of B.

Does that make sense?

And of course I mean you can spend time to calculate what is this and you know the area is I mean this is two two and then the

height is 2<unk> 3 and therefore total area is 12 * 4 * the base times the

height. Okay and that's 4<unk> 3. But

height. Okay and that's 4<unk> 3. But

even without doing this um we can already root out all other answers.

Does that make sense? That's how you can do it quickly.

Um, are there any questions, comments?

Uh, online people, >> I don't have a question.

>> Okay, good. Alex, did that make sense to you?

And Jolene, is that how you did it?

>> Yeah. Yeah.

>> Okay, good.

Um, yeah. So, so I think you you learn something and uh and some often times, you know, you don't have to do the all the calculations, right? That would save

you some time if you can figure out what are the key things. Um

and and when you have time you can you can you know calculate the rest but but I'm I'm hoping you can learn some uh useful techniques that have you have you save time and also broaden your

perspectives when you you know are facing a a problem you can think of multiple ways to solve it.

All right. So that's it for today. Thank

you for your attention and hope to see you again next week. Thank

>> you're welcome. Hi.

>> Bye.

>> Hi.

>> Bye, Michael. An next.