Proportion is used to show how quantities and amounts are related to each other. The amount that quantities change in relation to each other is governed by proportion rules.
Direct proportion is when one value of 2 is a multiple of the other. As an example; 1 cm equals 10 mm. In order to convert mm into cm you ALWAYS multiply by 10.
If one value is inversely proportional to another then it is written using the proportionality symbol in a different way. Inverse proportion occurs when one value increases and the other decreases. For example, more workers on a job would reduce the time to complete the task. They are inversely proportional.
Okay, we're looking at direct and inverse proportion. So we'll first look at what is proportion, what do we mean by it, what's the point of it? We'll do some subtle picks and then we'll finish off with some exam style questions. So you've why do we care about proportion? What's the point of learning about proportion?
So this is very good. If we're talking about maybe scale factors, we may be given some information and we need to kind of glean some extra information from what we've got already. Let's do an example. Let's say our friend gets back, there's a mango sales on the beach. We're sat on the beach and all the way down the other end, there's a guy selling mangoes.
Our friend gets back from the mango store and he says he spent £7, $50 on mangoes, and the guy gave him six mangoes. Well, let's say we've got, say, £13 and we need to buy eleven mangoes. So is £13 going to get us eleven mangoes? Is that going to be enough to buy eleven mangoes? Well, we can use vortality to work this out, right?
This is our problem. We don't want to waste a whole 40 minutes round trip walking down to the other end of the beach and back again if we can't afford the mangoes that we need. So we can use this bit of maths to save us a bit of hassle if we can't afford it. So let's jump in, right, let's say if six mangoes, 50. Well, to get one mango, we can scale this down by just dividing it by six, right?
We get from six mangoes to one mango. And what we do to the right hand side, we have to do the left hand side. We divide that by six as well. So 750 divided by six, that's going to give us six to 750, goes one, remainder 1.5, which is a quarter of six. So it's going to be one and a quarter.
It's going to be £1.25. That's the cost of one mango. We can now scale up, right? We can go from one mango up to eleven mangoes. So I can times by eleven.
And if we do the same to that side, we'll have to time set by that as well. 125 times eleven is going to give us twelve pound 50 plus 125, that's going to give us £13.75. That's how much it would cost for eleven mangoes. And so we can save ourselves a little hassle, because we know we're going to be 75ft short if we chase it all the way down the beach. And we can't afford the mangoes out of the brand.
And so we can use, fortunately, to save ourselves all that time and instead goes to the ATM before getting our mangoes. Right, let's look at some length, area and volume stuff. Again, if we're talking about proportion, this is very difficult here. So if we've got a kind of scalpactor between our two shapes. Let's say we've got two containers and one container is maybe one by one by one metre, and the other container is five x five x five metres.
Well, we can use our knowledge of proportionality to instantly work out how much more liquid we need to fill that big container. And in a simplest form, if we just look at three lines, right, this is maybe a one metre line, a two metre line and a three metre line. Well, the scale factors between these lines is we're timesing by two here and we're timing by three here. Very simple. Well, let's look at how this scale factor, this linear scale factor, that affects the area.
If we now imagine we've got an area, it's going to be the base of our box, and we're looking at how the area is going to be affected by this scaling up. Well, the area of our first box is just going to be one metre squared, right? It's just one x one, so it's one squared. The second box is going to be two by two. So it's going to be two times two or two squared.
And the arrow of the third box is going to be three by three, where it's going to be three times three. Or we can just do that as three squared. And if we look at the cubes next, if we look at how the cubes are affected by the scale factor, well, the first cube is going to be a one by one by one, so it's going to have just a volume of one cube or just one. Our two by two by two cube is going to have a volume of T times T times T. It's going to be length times depth times height, or in other words, it's going to be two cubed.
And our three by three by three cube is going to have a volume of three times three times three, which I'll write as three cubes. What we can see happening here, hopefully this is going to make something a second, is our linear scale factor is two and three. Right? In these two cases, our area scale factor is two squared and three squared. Our volume scale factor is two Cubs and three cubed.
And so whenever we go up a dimension, we end up adding another power onto our original linear scale factor. So that's how scale factor and proportion is related as we kind of go through the dimensions. Right, let's look at direct verse. Direct verse, inverse proportion. So how are these related?
How are they different from one another? Well, director is quite nice, right? Directors are very simple. If you've got two values, let's call them X and Y for simplicity. If they're directly proportional, it just looks like that or kind of like that.
Anything we want, really, it goes through zero. And if we double X, we double Y, right? If we pick this, if X is say three here. If we double X to give us six, then the Y values will also double. And same with this one.
If we pick a random point there and we double it to get to that point there again, y will double what will go from that value there up to that value there. So all this is saying is, this is relatively simple, right? We've got a straight line, our standard straight line equation probably from year nine is just Y equals MX plus C. If you want y equals MX plus C, well, C is the way intercept C is going to be zero. In this case, all of our direct proportional graphs happen on going through zero.
And so we're left with just Y equals MX. And that is really our equation for direct proportionality, right? We normally replace N with K, just some other random letter, doesn't really mean anything, but that's what we use for if we say Y is directly proportional to X, we can say this is our equation between Y and X, and that K is just going to represent the gradient of that graph. Inverse proportionality is slightly different. Inverse proportionality says if we double X, y is actually going to have and if we triple X, y is going to third.
So if we start with a very simple example, let's start off with just a random point. Let's say we've got X is one, Y is one. Well, if we double X, if X becomes T, y is going to half, y is going to go down to a half.
And if we quadruple X, if you go from one up to four, or Y is going to become a quarter. And likewise, if we have X, y is going to double. And if we quarter X, y is going to multiply by four. And so we end up with this curve that looks like this. And that curve is the kind of general curve of our inverse proportionality.
In this case, it's the graph of Y is equal to one over X. But in general, just like when we have a K here, this value doesn't need to be one necessarily. In general, we can say this is just some number over X, right? This number on the top is just going to be our scale factor between decide. If K was two, for example, every single value on this graph would just times by two, right?
That point there would jump up to that point there, that point there would go to there, that point there would go to there, that point there would go to there, and we end up with a new graph that looks like this. But it's the same shape, right? It's just moved up or down depending on the value of K. That's our inverse proportionality, y is equal to K over x. That's our formula for that, right?
Exponential graphs, exponential proportions. So this is where we've got x in the exponent. So x is now a power. So in simplest example, let's say we've got a virus and it's doubling every few days, right? So we could say if you've got Y and X here, it could be the number of people infected and x could be the number of days.
This could be two power X, right? So every day our virus is doubling in this case. So at the very start, anything to the zero is just going to be one. So we start off at one. After one day we've got T to the power of one.
That's two. After two days it's T to the power of two, which is two squared, which is four. After three days we've got two to power three. So it's three two cubed, which is eight. And you can see we've already kind of almost jumped off of there.
This is our exponential model. In this case, we generally have something that we get this like this almost like a kind of skateboard ramp. As we go along, further, further, further, the gradient gets increasingly steep. And again, this is just one example, right? This is just two to powerbacks.
In general, we can have anything. In the case of two, we generally use some value of K of x, or it can be A or P or whatever letter we want. That's our general exponential proportion. There's one more thing that we can look at, and this is all of these graphs. When X is zero, it's always going to be one, right?
Anything to the zeros one. So we may want to use the same principle we've used before and have a scale factor in front of it. We could look at say Y is equal to some scale factor a times K to the x. And that's going to give us basically every single point on this graph is now going to be multiplied by A. It's going to jump up to there and to there and there and so on.
So it's going to get scaled up like that. So that's our kind of general, all encompassing exponential equation. Let's finish off with a couple of example questions. So x is inversely portional to the square root of Y when X is twelve, y is nine. Find the value of x when Y is 81.
So we can start with our equation. It's going to be K over root Y. When X is twelve, y is nine. So we can plug in our values. We know twelve is equal to K over root nine, which is just K over three.
So we can rearrange this and we can say K is equal to 36. And so we can now get our full equation. X is equal to 36 over root Y. And if Y is 81, we end up with 36 over 81 is just nine. So we get 36 over nine, which is just four.
Saturday anything else? Let's do this one as well. If you've got two to five grammes of flour to make nine cakes, we want to make 20 cakes. Do we have enough flour? Well, we could do similar to the mango problem.
Right. We can work out how much flour is required for one cake. If two to five grammes is equal to nine cakes, then if we want just one cake, we know we've got to divide both sides by nine.
That's going to give us 25 grammes. I'd say 25 grammes is required for one cake. 20 cakes is going to be 20 times 25 to end up with 500 grammes needed. And it's she only has 475 and so she does not have enough power at the end. I'll.
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