Good afternoon. Welcome to Math Kangaroo Level 5-6 Webinar Number 4. Welcome. I'm Dr. Sarah Sagie. I'm the instructor for this webinar series for Level 5-6. If this is your first time joining us, welcome. I will start with a warmup problem because a lot of students join at this time. And then I'll go back to introducing myself in the lesson. So today's lesson is about ratio and proportion. Our warmup problem is a motorcyclist drove a distance of 28 kilometers in 30 minutes. How many kilometers would he drive in one hour if he drives at the same speed? I do have polls for some of the questions. This is a question where I have a poll today. If the polls interfere with your ability to read the problems, let me know in the chat. You can chat with me. Do not chat with the tablet co-host, but chat with me, Sarah Sagie. Now the poll should be launched. You should be able to put your answer in the poll. How many kilometers would he drive in one hour if he drives at the same speed? All right, most of you have answered. I will let you know Math Kangaroo does not deduct for wrong answers. So if you are taking a Math Kangaroo contest and you're not sure of the answer, either you think it's one of two answers or you don't even, you have no idea whatsoever, it's always better to guess on our particular contest. It's better to guess than to leave a blank, okay? I'm ending the poll here. I'll share the results. Most of you think it's 56 kilometers. I'm guessing that 58 was just a quick little arithmetic error that somebody made. If the poll does not close on your screen, go ahead and tap on the X or the word close that you have right now. So a motorcyclist drove 28 kilometers in 30 minutes, and we want to know how far they can drive at the same speed if they're going for one hour, and one hour is also 60 minutes. So we need to fill this in. Well, this was times 2, so this will also be times 2. 28 times 2 to make the equivalent fraction is going to be 56 kilometers. So the correct answer is B, which is what most of you said in the poll. So very good job. So obviously, as you figured out, today's topic is going to be about proportions and ratios. Hopefully you've had some introduction to this. Most of the proportions and ratios we'll use today, you'll notice I like to solve using fraction notation. However, you can't write ratios in other ways. You can write them as words. A pancake recipe calls for two cups of water to one cup of pancake mix. So that is a two to one cup to cup ratio of pancake for water to pancake mix. You can write this as a fraction, two cups of water, one cup of mix, or you can use a colon. So all of those work. Now, if I wrote this upside down, if I wrote 1 over 2 and my ratio was water to mix, that would be exactly backwards. It would be the reciprocal. It would be wrong. Keeping the correct numbers and the correct labels together makes a difference in these. Okay? All right. So I like to write the first number as the numerator and the second number as the denominator. So 3 to 5 is 3 fifths. When we talk about a ratio in lowest terms, that is the same as a fraction in lowest terms. So you can see an example. Here's a ratio of 6 to 7. We notice that each of these can be divided by 3. If we divide them by 3, we get 2 to 3 or 2 thirds. So you can reduce your ratios the same way you reduce fractions. Hopefully that's familiar to you. The precaution is when we state that two ratios or rates are equal to each other. A rate is a ratio where one of the things is time. So how many minutes, how long it takes you to do something or other units of time per second, like miles per hour, how fast you go in a car, that is a rate. But it is really a ratio as well. Right? So hopefully you are familiar with these terms. If not, now's a good time. A rate is just a ratio where part of it is time. So we can write a proportion as two fractions that equal each other. And a lot of you will have experienced in class making equivalent fractions, right? So I could say that 2 thirds is the same as 4 sixths. It's an equivalent fraction to 3, to 3 over, I have to multiply this by 1.5, 4.5, right? So we can always do equivalent fractions. Right? If the equation is true, then the ratios are equivalent. Now this is a trick that we use for solving proportions. If one of these is not known, this is a great trick. And it comes basically from algebra. So I'll show you what we do, and then I'll give you a little bit of an explanation about where it comes from. So if we have the proportion A is to B as C is to D, I can do a cross product. Product means multiply, right? Multiply. So I'm going to multiply A times D. When I don't have anything in there, it's times. And B times C. If those products are equal, then it was a true proportion. Let me explain where this comes from. If I have this proportion up above, and I multiply both sides of the equation by the same number, this is some basic algebra, and we do use algebra in level 5.6. So if we multiply both sides by B, you'll notice these cancel. And I end up with A equals CB over D. Now if I multiply both sides by D, I'll notice that this cancels, and now I get D times A equals C times B, which is exactly what we got here. So this cross-multiplying really is multiplying by both denominators. So in algebraic world, it works because we're multiplying by both denominators. If you like shortcuts, just think multiplying an X across. Everyone likes shortcuts. All right. So today I have polls for 1, 2, 3, 4 more of the questions, and I don't have polls for others. The reason is this group isn't so terribly large, so please feel free to put your responses to the problems or any questions in the chat. I will be glancing there as I go, and if I see questions, I'll do my best to try to answer them. Okay? The polls take a little more time, and because I want to be able to solve as many problems with you as possible, I don't use a poll for every single question. I also do not have our TA with us today, so if you see me taking some sips of water, it's pretty hard to talk for a whole hour without any water. In the time it takes Peter to solve two problems for the math kangaroo competition, Nick manages to solve three. In total, the boy solved 30 problems. How many more problems did Nick solve than Peter? This is a polled question, but I'll wait a few moments before I launch it, just in case you're one of the people where it blocks your screen. Do have some answers coming through. I still have some people asking for hints in the chat. That's great. I'm happy to help when I can. When I have that quiet time is usually when I try to get to reading the chat. So here's the poll. Remember, in a math kangaroo contest, go ahead and guess. If you don't see the poll on your screen, it might have to do with your settings. Don't worry about it. You can just put your answers in the chat. So on a real math kangaroo, do not leave blanks because we don't subtract anything and you might guess right. And if you might guess right, that's still points, right? All right, I'm gonna end the poll and share those results with you. The majority of students, about 70%, say six, but we've got some other answers. Let's take a quick try to solve this together. So if your screen doesn't automatically close the poll, close it. I have to close it on my tablet, but my computer closes all by itself. So I'm not sure which way your system is set up. Okay, in the time it takes Peter to solve two, Nick manages to solve three. So all together, together, they solved five, right? And we wanna know how many more problems did Nick solve than Peter if they solved 30 all together? Well, if they solved 30 all together, there's a few ways we could calculate it. I could say, okay, so five times what number equals 30? Five times six equals 30. So it would be Peter would have two times six, and Nick would have three times six. I can go ahead and calculate them, or I can just look at the numbers and say, oh, well, if it's two times six or three times six, then the difference is a six. So Nick was able to solve six more problems. If you want to solve it using the proportion way, you could say, all right, maybe Nick does three out of five, and the total is 30. So I have an unknown up here. Now, when I cross multiply, I get three times 30 is 90 equals five times X. I divide both sides by five, and I'm going to get 90 divided by five is 18 equals X. So 18 was the amount that Nick solved. I got the same answer. And if I do 30 minus 18, I'll get that 12 is the amount that Peter solved. So multiple ways of doing it. One of the things I love about having multiple ways to do a problem is you can check your work. If I get the same answer and I've used a couple of methods, the likelihood that I'm correct is very, very high. And if you solve it one way and somebody sitting next to you solves it a different way and you get the same answer, you can share with each other, learn from each other, and both have learned something new and get correct answers. I do have a poll for this one, but again, I'll give you some time. And if you don't see the polls or need a hint, let me know. There are cats and dogs in a room. The number of cat paws in this room is twice the number of dog noses. The number of cats is, Notice, we don't have an absolute concrete number, we just have a relative information here. Twice the number of dogs, equal to the number of dogs, half the number of dogs, one-fourth the number of dogs, and one-sixth the number of dogs. Pay attention that we are looking at cat paws, dog noses, and then just cats and dogs in the solution. All right, here's your poll and if you don't see the poll or you don't like, if you don't see the poll, don't worry about it. If your screen went blank, I'm sorry, hopefully you took some notes before it went blank. That's why I gave you some time. If you don't like the poll, you can close it right away, I guess. Great participation in the poll. Thank you all of you who replied. It's almost everyone, so I'll close it and share. Look at this, we have quite a few different results. The most common result is half the number of dogs. It's a very interesting answer. This is a number nine question, so it's one that we hope you will be able to do. All right, so it says the number of cat paws in this room. Hopefully, I'm not surprising any of you when I say your most cats have four paws, right? So we will assume that it's four paws on the cat. So four paws per cat. It says that the number of cat paws is twice the number of dog noses. So if I have four times the number of cats, and I compare that to the number of dog noses, every dog has only one nose, right? So it's one times D for dog. So that is, it says the number of cat noses is twice, the number of cat, ah, cat paws is twice the dog noses. So it looks like that, okay? So cat paws is twice dog noses. So how can we do this? The number of cats is twice the number of dogs. Then we would have a two and a one, and we would have an eight to one ratio, that is obviously not going to work. If we have an equal number of cats and dogs, then we would have a four to one ratio. If we have half the number of dogs, that would mean the number of cats is half the dogs, that means that dogs is twice the number of cats. Let's see how that works. Then we have four to two, and four to two does equal two to one. So the correct answer in this case is going to be that you have half the number, you have twice as many dogs, or, yeah, twice as many dogs, so half the number, the cats is half the dogs. When you start talking like this double and half, it gets my little tongue tied, it starts to show up. Okay, but the cats is half the dogs, or the dogs is twice the cats, that would be the same ratio. Okay, do not have a poll for this one because it comes hard to see the figures, right? In which of the four squares is the ratio of the black area to the white area the largest? Put your answers in the chat, I think this one should be a pretty fast one. Okay, very nice, I'm seeing a lot of answers in the chat. I'm going to solve this in two ways. So the first way I'm going to do is I'm going to find the ratio of black to, I think I'll do total in this case, okay? So if I look at this, I can see that this is one fourth of the square, right? Because if I divided it up into four pieces, this is one fourth, right? So that's one fourth. I can take a look at B, and I could imagine that this piece is moved over here, and this is shaded in, so this also is going to give me one fourth. Now let's take a look at C. I'm going to add a line through here, I'm going to bisect those, and now I have four out of one, two, three, four, five, six, seven, eight, four out of 16, four out of 16 is also one fourth, and I can do the same thing with D, and I can get that it's four out of 16, and that equals one fourth. So that gives me E, they're all the same. So the other method is kind of this one where I showed you, I can move this black and move it over here. So I could kind of redraw all of these. If you like to draw for your solutions, you might find that, okay, so if I draw this triangle shaded in over here, this triangle shaded in over here, and this triangle shaded in over here, I can make all of my pictures look the same just by moving the shaded regions until they're all up into the same corner, and they'd all look like A. The answer's E, all the same. Or go ahead and count your fractions out and see if you can make equivalent fractions of one fourth. There is a poll for this one, but I'll give you a few moments to solve after I read it, okay? A stick that in reality measures one meter is two centimeters long in a certain picture. And in that same picture, the height of the fence is 4.5 centimeters. What is the actual height of the fence in centimeters? Anybody knows what, when we have a picture of a representation of something, and we use a certain measurement, and we say that it's equivalent to a real-life measurement, if anyone knows what that is called, you frequently see it on maps, put that in the chat. I can't give you a bonus point, but you can impress me if you know what that's called. Make some notes if you are one of the people who can't see your screen, because I will launch the poll soon. So if you need to take a couple of notes down so you can keep working, do that. This group is pretty fast in answering the polls. Thank you. Here are the results. Interesting things is the most popular answers were 225 centimeters and 22.5 centimeters. Those are only off by what we call a factor of 10, right? Multiplying by 10 to go from 22.5 to 25 to 225. So let's see which way it needs to be. Let's take a look, close up your polls if needed. All right, so a student asked me for a hint and I said, you know, for me this was easier if I used all the same units. So I'm gonna change this one meter to be 100 centimeters. So now what I have is in real life, 100 centimeters is gonna be two centimeters on the drawing. And I can even simplify that to 50 centimeters in real life would be one centimeter on the drawing. And a student told me correctly that this is the scale. Right, so on a map, we would call it scale. We obviously can't draw maps to real life because how would you draw a map of your neighborhood to real life size? You'd have to fold it all up inside your house somehow. Right? Okay, so we're gonna use this and we're gonna set up a proportion 50 centimeters in real life, one centimeter on our picture. And we know that the fence is 4.5 centimeters in our picture and we don't know it's real life measurement. So when we do this multiplication, 50 times four is 200 and half of 50 is 25. So 225 equals one times X. So X and all of our units here were centimeters, right? So X is 225 centimeters. So the trick that I told the student is to go ahead and convert and get convert your units. One meter is 100 centimeters. Number five, the dog is nine times as heavy as the cat. The mouse is 20 times lighter than the cat. And the turnip is six times as heavy as the mouse. How many times is the dog as heavy as the turnip? There's no poll, so just quiet working time and you can put your answers in the chat. I'm seeing some answers in the chat, thank you. I'm going to set this up using a method that's called dimensional analysis. Okay? Math Kangaroo doesn't teach dimensional analysis, but I do, and you might use this in the future. This is the kind of thing you use when you set up something that is equal to 1. So in the previous problem, we actually did this a little bit when I said that 100 centimeters is the same as 1 meter. 100 centimeters and 1 meter are exactly the same length, right? So if I multiply something by that, I'm not changing its dimension. It's the same as multiplying by 1, the identity property. So think about that for a minute when I try to explain how I do this. So I'm going to set up some analyses. My units in this case are the animals, right? So it says the dog is 9 times as heavy as the cat. So 1 dog equals 9 cats, right? I'll switch colors. The mouse is 20 times lighter than the cat. 1 cat equals 20 mice. If the mouse is 20 times lighter, you're going to need 20 mice to be 1 cat in terms of weight. Obviously, they're different. Well, now I have a turnip. It's 6 times as heavy as the mouse. So I'm going to need 6 mice to be 1 turnip. Okay? I want to compare the dog to the turnip. So I want to compare the dog to the turnip. I want to have dog and turnip in my ending. Remember, each one of these is 1. So I can multiply things that are by 1. It's no problem. I want to have dog and turnip. How many of you love when you multiply fractions, when you find the common fractions and you put the common factors in the fractions and you cross them out? Right? So if I told you I did 9 over 12 times 3 sixths, how many of you would say, okay, well, this is the same as 1 half. And or let's do it crosswise. It'll be a little more like what we're doing. Let's say, okay, well, 3 goes into 12 four times. So I can do that. And 3 goes in here twice and 3 goes into 9 three times. So I can simplify this down and I get 3 over 8. How many of you love to do that? Right? You can do that. So this is the example with numbers. Yeah, I'm getting some people raising hands. Fine. This is the example with numbers. If you can do that with numbers, I'm going to tell you you can do that with units as well. So I have cats here and cats here. I can cross them out. I have mice here and mice here. I can cross them out. This is the process is called dimensional analysis. Okay? I'm left with only dogs and turnips. Now I multiply cross and multiply my numerators 1. Oh, I can simplify if I want. I could do that. I could say that 6 and 20 are both divisible by 2. So I have 3 and I have 10. And then I can do the 3 and the 9 and I can say that that's a 1 and this is 3. Now I multiply across. I get 1 over 3 times 10 is 30. So I need 30 turnips for every dog. How many times is the dog as heavy as the turnip? If I need 30 turnips for every dog, the dog is 30 times as heavy as the turnip. So now you've seen a little dimensional analysis. You've seen crossing and simplifying factors. That's one way to make ratio and proportion problems easier to calculate. So hopefully those are tools that you'll be able to use over and over and over again as you work these problems. Number 6. The length of a certain rectangle is 80 centimeters and its area is 3,200 centimeters squared. Find the length of another rectangle if its area and width are half the area and width of the rectangle described. If anybody doesn't remember how to calculate the area of a rectangle, let me know in the chat and I'll give you the hint. But I think you should know the areas of rectangles. Anybody else want to answer in the poll? Remember, the poll answers are anonymous, right? So I don't know what you put in the poll. If you make a mistake, that's all right. Some of you can keep track of how many times I say the wrong thing during a webinar. It's probably quite a few times. All right, I'll end the poll and share. Okay, we have a lot of you saying it's 40 and a lot of you saying it's 80 centimeters. Let's take a look. This is one where I still, I always like drawing. I'm kind of addicted to my drawings. Everyone is different, but I love my drawings. So I have a certain rectangle and then I have another rectangle and its width is half. So I still need to know the length, but I know its width is half. So how's that? So here's my length, here's my width. Here is one half of the width. Right? And I know that the area here is 3,200 centimeters squared. And I know that half must be 1,600. I was gonna say 1,600, which is another way to say it. Okay, so area equals length times width. If the width is half, then the area is also half even when the length stays the same. So we have the same length, it has not changed. If I was to go ahead and make the length half as long, right, then I would only have one quarter of the area because I would have area equals one half length times one half width. And that equals one fourth of length times width. So one quarter of the area. So when you change one dimension, you change the area by the same proportion. Okay, together, Adam, Bart, and Charlie earned $280 during their vacation. Adam made twice as much money as Bart and four times as much as Charlie. How many dollars did Charlie earn? Hey, I will start working on it. There is no poll. So if you're still working, you can keep going or you can see how I'm going to do it. So we have together that Adam, Bart and Charlie earned $280. Adam made twice as much money as Bart. So I'm gonna use a little algebra here. Adam equals two times Bart, just to make some notes, right? And Adam made four times Charlie. So by that logic, Bart has to be two times Charlie, right? So now we have some numbers here. So we know all together is gonna be Adam plus Bart plus Charlie equals $280. Charlie is the lowest earning one. So we'll just make him Charlie. He's also the one we want to know. We know that Adam was four times Charlie and Bart was two times Charlie. Adding that together, I have four plus two plus one is seven times Charlie equals 280. Divide by seven and Charlie equals $40 B. Now, how can I do that using ratios? It's a little bit different thinking than the algebra, but sorry, I meant to be erasing. Okay, so I'm gonna keep those ratios up here. So, if I want to use ratios, I know that Adam is 4 times Charlie, 2 times Bart. I want to know Charlie. There's another way to do it, which is I can make everything in terms of Adam, but these are going to be my ratios. So, Adam is 2 times Bart, and Adam is 4 times Charlie. And then Adam is, of course, himself. So I could do this, and I could add them that way as ratios as well, okay? So hopefully that helps you. I think this one works out a little bit better as an algebra problem, and we've done a little bit of algebra before. All right, so the ratio is the relationship between two numbers, and the proportion is kind of the relationship between two ratios. We know that they have to be equal to each other, right? So when we solve problems with ratios, it's helpful to use a fraction and simplify it. That tends to be the easiest way to calculate your ratios. It also allows you to use calculators if you were to be able to use a calculator, which you cannot on Math Kangaroo Exams. So one of the things you'll notice about the problems we work in Math Kangaroo Exams is that arithmetic part of it, the calculation is not too tricky. It's more the how do you set it up, what do you do? Now we're not done because I have some bonus problems. When we set up these lessons, we know that the webinars tend to go a little bit faster than the classes, and we add more problems to our webinars. So whenever we finish problems, I can keep going. The length on one side of the rectangle was increased by 10%, and the length of the other side of the rectangle was decreased by 10%. How did the area of the rectangle change? There is no pole, but I might encourage you to draw a rectangle. It doesn't say which one is which, but we can do length and width. One of them will increase, and one of them will decrease. Okay, so if I increase something by 10%, I now have 110% of the original, right? Or 1.1. I hope some of you have worked with percentages before, right? So if I increase by 10, I get 110, and if I write that as a decimal, it's 1.1. If I decrease something by 10, I now have 90% of the original, or 0.9. Area equals the new length times the width. So the original area was length times width. So this is original, and this is the new. The new area, area n for new, equals 1.1 times the length times 0.9 times the width. It doesn't matter which is width and which is length, right? It doesn't matter. I just put letters on the rectangle. I could switch it. Okay, so a n equals, I can rearrange this. I can do multiplication in any order because of the commutative property, right? So it's 1.1 times 0.9 times the length times the width. So I just have to figure out what this is. 1.1 times 0.99, and then how do I do my decimal points? There's two digits to the right of the decimal, so I need two digits to the right of the decimal here. So I get 0.99. So a of the new equals 0.99 of the original length times width, and this is 99%. Okay, now how does 99% match up with what I got to start with? It's not that it did not change. It's 99%, so it's less than. Nothing happened by 20%. It does not depend on the length of the sides. It does decrease by one percent because 100 minus one percent equals 99%. So the correct answer is B. Kind of a neat question. It doesn't matter what the sides are. Bonus question number two. A train traveling at a steady speed crossed a bridge which was 20 meters long in one minute. The whole train passed a person standing on the bridge in 12 seconds. How long was the train? I might, I do have a poll for this one, but we're going to run out of time, so I won't launch it, but you might try drawing a few pictures to help you with this problem. So, drawing does not have to be your specialty to do math kangaroo problems. You can see my drawing is very, very crude. And when I teach drawing lessons with math kangaroo, I always tell students, be very, very simple. Circles, rectangles, something real simple, right? Does my picture help? It's a very common math contest question to have a train crossing a bridge, and what you have to realize, when they talk about a train crossing the bridge, they talk about the front of the train to the back of the train, which means it's the length of the bridge plus the length of the train. I see several responses. All right, let's try this. What if I draw one more thing? What if I draw the train right here? That means that from here to here, the train has traveled exactly 200 meters. Right? And we know that it takes 10 seconds for a train to go from the beginning to the end at one point, like passing a person. So the train from here to here, instead of being one minute, it's gonna be one minute minus 12 seconds. So that's 60 minus, 60 seconds minus 12 seconds. Right? So 48 seconds is how long it takes the train to travel 200 meters. And we know that it takes the train 12 seconds to go its own length, X. Remember I said we can frequently simplify math kangaroo things. They don't make the calculations very difficult. We can just use equivalent fractions here. This is times four, right? Or divided by four. So we need to do the same thing here, also times four or divided by four, depending on if you're reading from right to left or left to right. So this would have to be 50 meters. See, the train is 50 meters long, since the rate of travel is the same. So hopefully you liked our rate and ratio problems. Proportions were fun, right? That one was a little bit extra tricky because we had to deal with how long is the train. So remember, one great way to get much, much better at math kangaroo problems is, of course, math kangaroo past exams. That's the thing that will be most similar. All of the questions we use in our webinars come from past exams. But if you wanna see all the types all at once, how many there are, 30 problems, how long do you take to solve each one, the best thing to do is to practice some past exams. And you will have discount codes in your registrations for that when you sign up for the math kangaroo contest, you'll also get codes for video solutions. I recommend taking a look at those. I hope you liked our lesson on ratios. We'll have a whole new topic next week and I'll see you then, happy Sunday.

ratios

proportions

Math Kangaroo

level 5-6

cross-multiplication

dimensional analysis

unit conversions

Dr. Sarah Sagie

equivalent ratios

problem solving