WEBVTT mathematics/basic-math/pyo
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Welcome back to Educator.com; for the next lesson, we are going to go over ratio.
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A ratio is when you compare two things.
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You are making a comparison between two quantities; it is also same as division.
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If you look here, there is three ways to express a ratio.
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There is three ways to write a ratio.
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If A is 1 and B is something else, you can say A to B.
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You can write it out, A to B.
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All this, whether you write it like this, like this, or like this, they are all read as A to B.
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But you can write it like this, like this using a colon or as a fraction A to B.
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You are still comparing A and B.
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For example, if I said what is the ratio, you are comparing boys to girls.
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Because I said boys first, boys is going to be written as A, the first one.
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Then girls has to be the second one; boys to girls.
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If I ask for the ratio of boys to girls, then I can't give you the number of girls to boys.
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You can't do this. You have to write out the ratio in the order that was asked for; boys to girls.
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If I say there are 5 boys and there are 3 girls, then the ratio of boys to girls would be 5 to 3.
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You can also write it as 5 to 3 like that.
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If I ask you what is the ratio of girls to boys, then you would have to
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give me this number first, the number of girls to the number of boys.
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You always have to write out the ratio in the order that it was asked for.
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A to B, A:B, and A/B as a fraction; this is called ratio.
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A rate is a ratio; you are still comparing A to B.
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But you are given different rates; for example, if I say miles per hour.
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Miles per hour would be... you have the number of miles and you have however many number of hours.
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You are comparing, you are making a ratio between the number of miles and the number of hours.
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A rate would be a ratio, same thing, A to B, but using different rates.
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If I say $5 for 5 candies, then that is a ratio.
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You are making the comparison between the amount over the number of candy.
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If you make a comparison between two things, it is called a ratio.
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When those two things have some kind of unit, then it is called a rate.
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A unit rate is a rate with a denominator of 1.
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That means that if I say I traveled 2 miles in 2 hours.
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Here is my ratio, 2 miles every 2 hours; this is my ratio.
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A unit rate would mean to make this, the denominator, the bottom number, a 1.
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That means I need to change this to become 1 hour.
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That would be a unit rate.
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That means in order to turn this 2 into a 1, I have to divide the 2.
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I am going to divide this by 2 which means I have to divide the top number by 2.
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This would be 1 mile per hour because it is 1 hour.
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1 mile per hour would be the unit rate.
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This alone would just be a rate.
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But when you make the denominator a 1, a unit of 1, then this is a unit rate.
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This here is a unit rate because the denominator is 1.
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Here is an example; $10 per 20 pieces; that is like the candy example.
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If it is $10 for every 20 pieces,
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in order to give me a unit rate, I want to find out how much it is per piece.
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One piece, I am turning this denominator into a 1.
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That means in order to turn this denominator into 1, I have to divide it by this number, divide it by itself.
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That means I have to divide this top number.
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Because this is money, I want to change it to a decimal.
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I know that 10 divided by 20 or 10 divided by 20 is going to give me 0.5.
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0.5 in money is the same thing as 50 cents.
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If I add a 0 here, that becomes 50 cents.
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Not 5 cents, be careful; this is 50 cents.
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The unit rate would be 50 cents per piece.
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I can put 1 in front of it if I want.
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But if I just say per piece, then I am talking about 1 piece.
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You can leave it like this; this would be your unit rate.
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When you are converting rates, rate remember it is a ratio of two different rates.
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You have different units on the top and the bottom; that is a rate.
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To convert rates means you are going to go from whatever rates they give you,
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whatever units they give you, and you are changing it to something else, changing it to different rates.
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You are converting them.
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For example, if I have miles per hour, let's say I want miles per hour.
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I am going to put miles on top; I am going to put... 1 mile per hour.
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This is the rate that I am starting off with.
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I want to convert it to feet per minute.
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This is miles; mi is miles; min is for minutes.
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I am going to convert this number here, this ratio, this rate, to this rate, this ratio.
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Remember rates are ratios; I am going to convert this to this.
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That means miles I need to change to feet and hours I am going to change to minutes.
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Miles and feet, they are both measurements of distance.
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Mile and feet, they are both measuring the distance of something.
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Hour and minutes, they are both measuring time.
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I can convert miles to feet and hours to minutes.
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In order for you to be able to convert rates, this to this, you have to know the equivalent units.
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How many feet equals a mile?--1 mile equals 5280 feet.
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This and this are the same; 1 mile is equal to 5280 feet.
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Same thing for hours and minutes; I know that 60 minutes equals 1 hour.
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If it helps, you need to just write this on the side.
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You are going to use this to help you convert rates.
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First thing I do, I am going to start here and I am going to end here.
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I am going to change all these into these.
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I am going to start from 1 mile to 1 hour.
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I am going to multiply it to different units because I can cross cancel things out.
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If I say that 5280 feet is the same thing as 1 mile, if they equal each other,
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then I can say 5280 feet over 1 mile is going to equal 1 because this number and this are the same thing.
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We said they are equal; this over this is equal to 1.
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Anything over itself is 1.
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If I said, for example, 5/5, isn't that 1?--because it is the same number over itself.
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Same thing here; this equals this.
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If I say 5280 feet over 5280 feet, isn't that equal to 1?
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This does equal 5280 feet.
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If I write it like this, you have to understand that this is the same thing as 1.
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If I multiply this by 1, I am not changing this.
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I can multiply this by 1 if I want because it doesn't change.
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Instead of multiplying it by 1, I want to multiply it by this.
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This is the same thing as 1.
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I am going to multiply all this to this.
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I want the miles to go away because the miles is going to have to change to feet.
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I need to the miles to go away.
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In order for me to cross cancel the miles, I have to have one on the top and one on the bottom.
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This miles is going to go down here.
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On the top, it is going to go 5280 feet.
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That way this and this will cancel.
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Again this whole thing is just equal to 1.
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I can just multiply it to this if I want.
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It is not going to change my answer because I am just multiplying it by 1.
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Same thing for hours.
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I also know that since this 60 minutes is equal to 1 hour,
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if I put 60 minutes and I divide it by 1 hour,
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since this whole thing equals this whole thing, this is also equal to 1, isn't it?
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They equal each other.
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Whenever the top and the bottom equal each other, that always equals 1.
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I want to multiply this whole thing to this whole thing because again this is equal to 1.
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I want the hours to go away.
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That means if this is already in the bottom, then I need to write this on the top.
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This is going to go 1 hour over 60 minutes.
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I just flip this; this went to the top; this went to the bottom.
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Because again if this is the same thing as this, then isn't this over this the same thing?
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I am writing it on the top and the bottom, depending on where I have to cancel it.
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If this is already in the bottom, then I need to cancel this.
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That is going to go like that.
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If I look on the top, what units am I left with?--feet.
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For my answer, if I multiply all this out, then I am going to be left with feet which is what I want.
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On the bottom, what am I left with?--minutes.
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That is what I want left on the bottom.
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I know that all I have to do is now solve this out.
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I cancelled out everything that I need to cancel out.
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If I just multiply this out and then multiply that out, solve for it, I will get my answer.
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Here my top is going to be 1 times 5280 times 1 which is 5280 feet over 60 minutes.
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1 mile per hour is the same thing as 5280 feet over 60 minutes.
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This would be my answer; I can simply this if I want.
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This is a ratio, is a rate.
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But if I want to change it to a unit rate, I can divide this by... let me use a different color.
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I can divide this by 60 and then divide this by 60.
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5280 divided by 60 is going to give me... I am going to cross out these 0s.
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Remember you can cross out the 0s if you want.
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It is going to give me... 8, 4, 8; it is going to give me 88.
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Again to multiply this, you are going to do 6 times 8.
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Write it out, 48; my remainder is 4.
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I am going to bring down this 8; then 6 times 8 is 48.
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My answer is 88 feet per minute; this would be my unit rate.
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Let's do a few more examples; the first example, write in simplest form.
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Here these are just ratios; it is comparing this number to this number.
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They look like fractions.
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But you can also think of these numbers, the top number and the bottom number as ratios.
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It is like division.
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To write this in simplest form, 12/36, I can look for a common factor.
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The greatest common factor is 12 because 12 goes into 12 here and 12 goes into 36.
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If you don't see that 12 is the biggest factor,
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you can just look for any factor because 12 and 36 have a lot of common factors.
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If you want, you can just divide the 2 first and then just keep making the numbers smaller.
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You can divide this by 4; divide it by 3.
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Since I know that 12 is my biggest factor, I am going to divide this by 12 and then divide this by 12.
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Whenever I am going to simplify, then I need to divide both
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the top number and the bottom number by the same number, the same factor.
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This is 1 over... 36 divided by 12 is 3.
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This is saying that the ratio of 12 to 36 is the same as 1 to 3.
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They are the same ratio; they are equivalent; they are the same.
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This next one, I know because this ends in a 0 and this ends in a 5, that they are both divisible by 5.
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I am going to take 30 divided by 5; 35 divided by 5.
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30 divided by 5 is 6; 35 divided by 5 is 7.
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This is simplest form.
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That means the ratio of 6 to 7 is the same as 30 to 35.
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Same thing here; let's divide this by...
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Again if you just see any common factor, you can just keep dividing until you get simplest form.
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Or if you find the greatest factor, that would be the fastest way.
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But let's say that we wanted to just divide this by 2 because I just noticed that they are both even.
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That is not the greatest factor; but let's just do that first.
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I am going to divide both the top and the bottom by 2.
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This is going to be 8/12.
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This is still not simplest form because they are both even still.
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4. or maybe 2 if you just notice that they are both even numbers.
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But from these two, the greatest factor is 4; let's just divide them by 4.
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8 divided by 4 is 2; over 3; that would be simplest form.
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For this, let's see, this is not an even number so I know that 2 is not going to go into them.
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If I add these two together, 5 plus 1, that is 6.
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6 is a multiple of 3.
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This bottom one, 1 plus 9, 9 is a multiple of 3.
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So I know that 3 can go into both of these.
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Divide this one by 3; divide this one by 3; 51 divided 3.
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If you don't know what that is, you can always just divide it.
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51 divided by 3; 1; subtract the number; bring this down.
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That will be 17 over... 18 divided by 3; 3 times what equals 18?
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That is 6; that would be simplest form.
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Next example, Tommy has 4 blue marbles, 3 green marbles, and 7 red marbles in a bag.
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Find the ratio of red to blue marbles; 4 blue, 3 green, 7 red.
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We want to find the ratio of red to blue.
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The ratio is going to be red to blue.
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How many red do we have?--7 to 4 blue.
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Make sure you have to write it as 7 to 4 and not 4 to 7.
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Because they ask for red first before the blue, you have to write out the red first.
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It is 7 to 4; you can say 7 to 4.
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Or you can say 7 to 4 like that as a fraction.
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Next, out of 27 students in classroom, 15 are boys.
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Find the ratio of boys to girls; ratio is boys to girls.
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They don't give us a number of girls.
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They just tell us that there is 15 boys.
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But I know that if there is 27 students total and 15 are boys, then the rest of the students have to be girls.
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I have to subtract 27 students minus 15; I am going to get 12.
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That means I know that 15 are boys and 12 are girls.
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The ratio of boys to girls would be 15 to 12 or 15/12.
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Find the unit rate.
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Remember unit rate is when you have a ratio and the bottom number, the denominator, has to be 1.
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Here the ratio is 250 for every 2 dozen.
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I want to find how much it is going to be for 1 dozen or how much per dozen.
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Think of it as per.
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Every time you see unit rate, you are going to think of per.
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Per whatever the unit is on the bottom; how much per dozen?
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That means I need to turn this into a 1.
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I divide this by 2 then to turn 2 divided by 2 into 1.
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Then I have to multiply the top by 2.
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250 divided by 2.... remember to bring out the decimal.
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It is going to be 1; 2; bring down the 5.
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2; 4; 1; bring down the 0; 5.
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Going to be $1.25 per dozen; there is my unit rate.
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A car goes 300 miles... mi means miles... on 10 gallons of gas.
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300 miles on 10 gallons of gas; find the unit rate.
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That means I want to turn this into 1.
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It is going to be how many miles per gallon.
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Then again divide this by 10; divide the top number by 10.
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300 divided by 10... every time you divide by a number with the 0 at the end of it,
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and they both have 0s at the end, you can just cross out one of the 0s.
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If I cross out this 0 and cross out this 0, then I am going to be left with 30.
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30 miles per gallon; this is the unit rate.
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A skydiver falls 240 feet in 5 seconds; 240 feet every 5 seconds.
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How many feet per second?--1 second; divided by 5; divided by 5.
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240... let's do it over here; 240 divided by 5 is going to be 4
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because that is going to be 20; 4, 0; that will be 8.
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48 feet per second; here is my unit rate.
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The fourth example, we are going to convert the units.
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This is going to be the most difficult part of this lesson.
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But just make sure you are going to...
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Just try to cancel out the units so that you end up with the units that you want for your answer.
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A car is moving at 8 miles per hour.
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I am going to write that as a fraction; 8 miles per hour.
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I want to convert this to feet on the top with what units on the bottom?--minutes.
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I know it is 10 minutes.
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But then I just want to focus on converting these units first--miles per hour to feet per minute.
00:27:37.100 --> 00:27:46.700
I am going to start off here; I am going to write that over 1 hour.
00:27:46.700 --> 00:27:53.100
Again I am going to multiply; I want to turn the miles into feet.
00:27:53.100 --> 00:28:05.400
On the side, let's find out... 5280 feet equals 1 mile; this equals this.
00:28:05.400 --> 00:28:18.000
That means if I put this over that as a numerator and denominator, it is going to equal 1.
00:28:18.000 --> 00:28:21.500
Why don't I just do that right now; I am going to do times...
00:28:21.500 --> 00:28:27.500
From the feet and the miles, which one do I want to go as my numerator?
00:28:27.500 --> 00:28:34.200
I want my miles to go on the denominator because I want them to cross cancel out.
00:28:34.200 --> 00:28:37.200
From these two, I am going to put this on the bottom.
00:28:37.200 --> 00:28:47.600
I am going to put 1 mile on the bottom and then 5280 feet as my numerator.
00:28:47.600 --> 00:28:53.000
That way my miles cancel out.
00:28:53.000 --> 00:29:03.700
Hours I am going to change to minutes; 1 hour equals 60 minutes.
00:29:03.700 --> 00:29:07.500
Write that out first so you can see it; it is a lot easier.
00:29:07.500 --> 00:29:11.700
Then from this and here, one is going to go on as my numerator.
00:29:11.700 --> 00:29:15.200
One is going to go on my denominator.
00:29:15.200 --> 00:29:18.600
Which one do I want to go on the top?
00:29:18.600 --> 00:29:20.600
The hours because here the hours is on the bottom.
00:29:20.600 --> 00:29:23.100
I want it to cancel out so I have to put it on the top.
00:29:23.100 --> 00:29:27.200
This is going to go on the top; this is going to go on the bottom.
00:29:27.200 --> 00:29:38.800
1 hour over 60 minutes; cross cancel that out.
00:29:38.800 --> 00:29:48.900
Here I want to now solve this out because if I look at the top, what units are left on the top?
00:29:48.900 --> 00:29:51.600
Feet is left which is what I want.
00:29:51.600 --> 00:29:57.000
This is what I want my answer in; I am on the right track.
00:29:57.000 --> 00:30:00.500
On the bottom, what do I want left?--minutes.
00:30:00.500 --> 00:30:02.900
That is where I am at; so I am good.
00:30:02.900 --> 00:30:09.200
Now I know I just have to multiply this out and solve these numbers out.
00:30:09.200 --> 00:30:13.400
Before we start multiplying this times this and get a big number and then
00:30:13.400 --> 00:30:20.000
have to divide by a big number, let's try to cross cancel some stuff out.
00:30:20.000 --> 00:30:24.900
Anytime you are multiplying numbers and you have numbers on top and you have numbers on the bottom,
00:30:24.900 --> 00:30:29.200
you can start cross cancelling things out if they have common factors.
00:30:29.200 --> 00:30:34.400
First thing I see is I see a 0 here and I see a 0 here.
00:30:34.400 --> 00:30:40.000
I can cross cancel those out.
00:30:40.000 --> 00:30:45.800
This is going to change to 6; this is going to change to 528.
00:30:45.800 --> 00:30:51.800
Cross out that 0; cross out that 0.
00:30:51.800 --> 00:30:57.900
8 and 6, I know that they have a common denominator of 2.
00:30:57.900 --> 00:31:04.800
I can cross this out, divide this by 2; I get 3; that changes to a 3.
00:31:04.800 --> 00:31:11.600
I am going to change that because that common factor was a 2 so that changes to a 4.
00:31:11.600 --> 00:31:15.400
That means I divided this by 2 and I divided this by 2.
00:31:15.400 --> 00:31:19.200
This became 4; this became 3.
00:31:19.200 --> 00:31:26.300
Here, does 528, is it divisible by 3?
00:31:26.300 --> 00:31:32.000
If you add this, it becomes 5 plus 2 is 7; plus 8 is 15.
00:31:32.000 --> 00:31:36.200
Is 15 a multiple of 3?--it is.
00:31:36.200 --> 00:31:39.600
Therefore I know that this is divisible by 3.
00:31:39.600 --> 00:31:45.500
If you are wondering what I just did, I used the divisibility rule.
00:31:45.500 --> 00:31:51.300
The divisibility rule of 3 is you add up all the digits.
00:31:51.300 --> 00:31:57.900
You are going to do 5 plus 2 plus 8 which gives you 15.
00:31:57.900 --> 00:32:02.000
You are going to see if that number is a multiple of 3.
00:32:02.000 --> 00:32:04.900
Does 3 go into that number?
00:32:04.900 --> 00:32:09.600
5 plus 2 plus 8 is 15; 3 does go into 15.
00:32:09.600 --> 00:32:20.300
I know that 3 will go into this number; 528 divided by 3.
00:32:20.300 --> 00:32:27.900
This is 1; this becomes 3; subtract it; you get 2; bring this 2 down.
00:32:27.900 --> 00:32:36.700
3 goes into 22 seven times; that is 21; 1; bring down the 8.
00:32:36.700 --> 00:32:59.400
3 times 6 is 18; this goes away; this became 176.
00:32:59.400 --> 00:33:03.100
Now all I have to do to find my answer is just...
00:33:03.100 --> 00:33:10.700
Since the bottom number is 1 times 1 and these all canceled out, then it is just 1 times 1.
00:33:10.700 --> 00:33:17.300
That is just 1 minute; I just multiplied all the numbers; I get this left.
00:33:17.300 --> 00:33:25.500
On my top, my numerator, it is just 4 times 176.
00:33:25.500 --> 00:33:57.500
That is 24; 7 times 4 is 28; add 2; this becomes feet.
00:33:57.500 --> 00:34:05.600
This is in a unit rate; 704 feet per minute.
00:34:05.600 --> 00:34:11.300
I want to know how many feet it will move in 10 minutes.
00:34:11.300 --> 00:34:19.000
What does that mean?--this is my unit rate; I am converting the units.
00:34:19.000 --> 00:34:21.400
This would be the correct answer.
00:34:21.400 --> 00:34:29.300
But then it is asking me how many feet it will move in 10 minutes, not per minute.
00:34:29.300 --> 00:34:34.100
If they asked how many feet it will move per minute or in one minute,
00:34:34.100 --> 00:34:39.500
this would be my answer, 704 feet per minute, for one minute.
00:34:39.500 --> 00:34:45.900
But since they are asking for 10 minutes, I need to change my denominator to a 10.
00:34:45.900 --> 00:34:48.600
They are not asking for a unit rate.
00:34:48.600 --> 00:34:52.800
They want to know how many feet for 10 minutes.
00:34:52.800 --> 00:34:54.400
I need to change this 1 to a 10.
00:34:54.400 --> 00:34:58.500
In order to do that, I have to multiply by 10.
00:34:58.500 --> 00:35:02.300
Same thing here; I need to multiply by 10.
00:35:02.300 --> 00:35:05.700
If I need to multiply this by 10, I just have to add a 0 at the end of it.
00:35:05.700 --> 00:35:06.500
That is pretty easy.
00:35:06.500 --> 00:35:18.600
It just becomes 7 thousand, 0, 4, add the 0, feet per 10 minutes.
00:35:18.600 --> 00:35:27.700
My answer, how many feet?--it is 7040 feet.
00:35:27.700 --> 00:35:31.000
I know that problem seemed a little bit complicated.
00:35:31.000 --> 00:35:39.300
But all I had to do was convert the units at miles per hour to feet per minute which is what I did.
00:35:39.300 --> 00:35:43.400
Multiply your top numbers across; multiply your bottom numbers across.
00:35:43.400 --> 00:35:46.600
If you want, you didn't have to cross cancel all this stuff out.
00:35:46.600 --> 00:35:50.600
That is why it looks so complicated, because we ended up cross cancelling numbers out.
00:35:50.600 --> 00:35:52.800
But if you want, forget about the cross cancelling.
00:35:52.800 --> 00:35:58.500
Just multiply all the numbers straight across; get this number.
00:35:58.500 --> 00:36:00.900
Multiply all the bottom numbers straight across and get this number.
00:36:00.900 --> 00:36:05.400
Then simplify if you want; you can do it that way.
00:36:05.400 --> 00:36:09.900
Once we get this, this is per minute; denominator is 1.
00:36:09.900 --> 00:36:12.700
This is our unit rate.
00:36:12.700 --> 00:36:14.300
But then because they are asking for 10 minutes,
00:36:14.300 --> 00:36:19.000
I need to change this denominator to 10 by multiplying by 10.
00:36:19.000 --> 00:36:27.600
We multiply the top by 10; you get 7040 feet per 10 minutes.
00:36:27.600 --> 00:36:34.500
Let's try one more problem; the sprinklers used 2 gallons per minute.
00:36:34.500 --> 00:36:42.400
How many quarts will it use in 30 seconds?--again we have to convert units.
00:36:42.400 --> 00:36:49.500
This is 2 gallons per minutes.
00:36:49.500 --> 00:36:59.600
I want to convert this to quarts... this is quarts... per seconds.
00:36:59.600 --> 00:37:02.800
I am going to put just 30 seconds here.
00:37:02.800 --> 00:37:06.500
Question mark, how many quarters per 30 seconds?
00:37:06.500 --> 00:37:17.200
I am going to start off with this again; 2 gallons per minute.
00:37:17.200 --> 00:37:34.300
Since I need to convert gallons to quarts, I know that 1 gallon is equal to 4 quarts.
00:37:34.300 --> 00:37:41.700
Remember if this is equal to this, I can change this to a fraction, 4 quarts over 1 gallon.
00:37:41.700 --> 00:37:43.500
That is going to equal 1.
00:37:43.500 --> 00:37:49.400
1 gallon over 4 quarts, that is also going to be the same.
00:37:49.400 --> 00:37:55.000
I can multiply this by... what do I want to get rid of?
00:37:55.000 --> 00:37:59.700
I want to get rid of the gallons first by using this.
00:37:59.700 --> 00:38:02.300
The gallons is going to go on the bottom.
00:38:02.300 --> 00:38:07.600
This one is going to go on the bottom; that way this will cancel like this.
00:38:07.600 --> 00:38:12.800
This one will go on the top like that.
00:38:12.800 --> 00:38:15.600
See how one goes on the bottom and one goes on the top?
00:38:15.600 --> 00:38:19.300
Or one goes on the top and one goes on the bottom?
00:38:19.300 --> 00:38:22.900
Just depends on what you have to cancel.
00:38:22.900 --> 00:38:30.000
Then I need to convert minutes to seconds.
00:38:30.000 --> 00:38:42.300
The minutes to seconds is going to be 1 minute is equal to 60 seconds.
00:38:42.300 --> 00:38:46.100
I need to write the minute one on the top so that it will cancel.
00:38:46.100 --> 00:38:50.200
This is going to go on the top; this is top, bottom.
00:38:50.200 --> 00:39:01.100
1 minute over 60 seconds; minutes will cancel.
00:39:01.100 --> 00:39:02.700
What units do I have left on the top?
00:39:02.700 --> 00:39:06.400
I have quarts which is what I want.
00:39:06.400 --> 00:39:09.700
And I have seconds on the bottom which is what I want.
00:39:09.700 --> 00:39:13.700
Now I just have to solve it out.
00:39:13.700 --> 00:39:19.400
If I want, I can cross cancel out this 2 and this 60.
00:39:19.400 --> 00:39:22.500
2 goes into 2; this changes to a 1.
00:39:22.500 --> 00:39:29.100
2 goes into 60; cut it in half; that is 30.
00:39:29.100 --> 00:39:30.700
You can cross cancel out again.
00:39:30.700 --> 00:39:44.200
But then otherwise you can just write it out; 4 quarts over 30 seconds.
00:39:44.200 --> 00:39:46.400
The reason why I decided to leave it...
00:39:46.400 --> 00:39:49.700
You could have cross cancelled it out; that is fine.
00:39:49.700 --> 00:39:53.600
Here they ask for 30 seconds.
00:39:53.600 --> 00:39:59.400
They want to know how many quarts will it use in 30 seconds.
00:39:59.400 --> 00:40:04.400
Here it will be 4 quarts every 30 seconds.
00:40:04.400 --> 00:40:09.800
I know that my answer will be 4; 4 quarts.
00:40:09.800 --> 00:40:17.400
It is 4 quarts per 30 seconds; that is my answer.
00:40:17.400 --> 00:40:21.000
That is it for this lesson; thank you for watching Educator.com.