Fill in the gaps: Two quantities a and b are said to be directly proportional if one quantity is always a constant ........ of the other.

Fill in the gaps: Two quantities x and y are said to be inversely proportional if x increases as y ............... at the same rate, so their product is constant.

Fill in the gaps: Two variable quantities are said to be ........................ proportional if the product of the two quantities is constant.

Three builders take 15 days to complete a project. How many days does would it take one builder?

It takes 4 friends 6 hours to clean the garden. How long will it take 2?

Inverse Proportion

﻿Hi there and welcome to another maths lesson, with me Dr. Sanda. In today's lesson we'll be looking at solving inverse proportion questions in context. You need a pen and a paper for this lesson. So if you did not have these handy, please pause the video and go and grab them. And when you are ready we can make a start. Cala fills a fish tank. She has a range of jugs that she can use to carry water to the tank. If Cala uses a four litre jug, she will need to use fifteen jug fulls. How many jug fulls are needed, if she uses a six litre jug. So I want you to think about this question, and think about if she changes the capacity of the jug, will she need more jug fulls or less? How would you go about solving this question? Please pause the video and spend about five minutes, thinking about to trying this task. Resume the video once you are finished. Welcome back. How did you get on with this task? Okay. So now we're going to look at various methods that we can use to help us solve questions like this. Our first step is always to identify, are we looking at a direct proportion or an inverse proportion question. Now, by looking at this question, we can see that if Cala uses four litre jugs, she will need to use 15 jug fulls. If she uses six litre jug, so now she's increasing the capacity of the jug. Well, she will need less jug fulls to, completely fill the fish tank. So higher capacity, she will need to use less jug fulls. She would have to fill it less times, therefore it's inversely proportion. So the first thing we can say is that the number of jug fulls is inversely proportional to the capacity of the jug that is being used. The product therefore will always be constant. So the number of jug fulls multiplied by the capacity, will always be a constant number. We discussed this last lesson where we drew the tables and we had the product X, Y. So fifteen multiplied by four is sixty. So we know that any combination that we use, the product has to always be sixty. So if she uses a six litre jug, it has to be six times something that gives me sixty. And that number is ten. We can use the inverse for this, so we can say, sixty divided by six is ten. So six multiplied by ten is sixty. So we know that she can use, a six litre jug and if she uses that she will need to do it ten times, she will need ten jug fulls. So Cala will need ten jug fulls, and we can answer the question like this. Now I'm going to show you a diagram here, and talk you through it. So here I have a grid, and what have showed you here, I drew a rectangle to represent that multiplication, the product of fifteen times four using area. Now, if you look at this first rectangle here, I have an area of sixty, because I want the product fifteen multiplied by four. Now I can represent this using different rectangle. So this time I had six litre jugs so I have six, and I need to find something that will give me exactly the same area. So the number of squares here is equal to the number of squares here, it's 60. I can also ask myself now, is there any other combination that I can use? As long as that area stays sixty, if I use any numbers here, this can be combination that she can use. So I can, for example, say, well, to get sixty, I need three multiplied by twenty. So now I can go and say, well, you know what, she can use a three litre jug, And if she does that she will need twenty jug fulls. I'm maintaining that product being constant, twenty times three. Okay now lets have a look at another method, for solving the question. The number of jug fulls N, is inversely proportional to the capacity of the jug used B in litres. Now, I have used N and B here, you could have used any different letters, as long as the question is not specifying loss. Now, what I know is that the product will always be a constant value. So N multiplied by B, is going to be a constant value. What is that value? So in the question we have been told, if Cala uses a four litre capacity, a jug that has a four litre capacity, then she will have to do it fifteen times, she will need fifteen jug fulls. So fifteen multiplied by four is sixty. So I know that the product of N multiplied by B, is going to be sixty. So I can write down N,B is equal to 60. Now the question asked me, if the capacity of the jug is six litres. What happens? How many jug fulls when she need? So I can say now, if B is equal to sixty, then N multiplied by six is equal to sixty, So I'm substituting here for B. Now I have an equation that I can solve N multiply by six equals sixty. To find that N , all I have to do is sixty divided by six. So am dividing both sides to the equation by six. And that gives me ten. So I got the same answer that I had from the previous method. So I know now that she needs 10 jug fulls, if she uses a six litre capacity jug. Now let's look at a slightly different method. That's our third method for now. I can write it down like this, number of jugs to the capacity. And if she uses four litres, she will need fifteen jugs. I can work out two litres first. I've divided by two. So what I need to do to the other side is, multiply by two so the fifteen becomes thirty. Then now that I know for two litres, I can work out for six litres. So that I've done six litres, by multiplying from two litres to six litres multiplied by three. So on the other side, I need to divide by three. So the thirty divided by three, and that will give me ten jugs. So I've ended up with the same answer using this method. You can see with this method, whatever I do to one part or one side, I do the inverse of the other. So here I divide by two on the other side I'll multiply by two. Multiply by three, I will divide by three. I found first two litres and that six litres, because I thought that would be easier for me, than having to move from four to six, because then it will involve a fraction. And it will involve me thinking, okay, how do I get from four to six? And I will need to use a fraction. And I wanted it to make things a little easier for myself, so I found two litres first. You could have found one first and then found six, or you could have done it in a single step. So now you have three different methods. So now you have three different methods, that you can use to solve questions like this. The first one involved us writing down the product or maybe drawing rectangles to represent the area and find the sides of the rectangles. The second method is actually write an equation. Write an equation where you have the product as the constant value, or you can write it down like we would solve, normal ratio or proportion questions. But be careful that if we multiply on one side, we do the opposite or the inverse on the other side. Out of these three methods, which one do you prefer? Okay really good. I prefer the algebra one. So I prefer writing an equation, with something constant when I'm representing the product, because I find that, that's really systematic and it's unlikely for me to make a mistake in that. But what I also like is to try and use two methods to double check that my answer is always correct. That's something that you can do. Now it's time for you to have a go at the independent task. You have three questions to answer. Please read each question carefully and decide whether the question is asking you, to use direct proportion or inverse proportion to solve the question. Once you've answered the question, please try and use a second method. So find another way of solving it, to check did you get the same answer using more than one method? If you do the question using two different methods and you end up with the same answer, then most likely your answer is correct. Please pause the video and spend between ten to fifteen minutes to complete the independent task. Resume the video, once you've finished. Welcome back. How did you go on with this task? Did you answer each question using at least two methods? Really good. Let's go through some of the solutions so you can mark and correct your work as we go along. The first question, Oliver, from Steel Construction Company, can build a house by himself in 48 days. If Oliver and Jane worked together to build the house of the same size, how long will it take to build the house? So I started with this, the people to time. One person takes 48 days, so two people double that will take half the time. So I divided by two and that was 24 days. So what I did to one side, which was multiplied by two, I did the inverse of the other and I divided by two. Did you have this method? Really good. Now the second method is, I said to myself, well, one multiplied by forty eight is forty eight. So two multiplied by what number gives me forty eight. And that number must be twenty four days. I know this, because I know that the product has to be forty eight, because there is an inverse proportion relationship here. Now the other method is I said, okay, you know what, I'm going to use algebra, so I'm going to write people, I'm going to use them as a P, for time I'm going to use the letter C. So I started by writing that down. So that I can write an equation, to represent what is happening. Also represent the product of PT is equal to 48, then I substituted it P is equal to two. So two T equals 48. To solve I need to divide both sides of the equation by two. And that gave me T is equal to 24. And you can see here with my three methods, I ended up with exactly the same answer. How many of these methods did you have? Really good. Let's move on that next question. So for question two, it takes six hours to fill a pool, with water coming from three taps. How long would it take if only two taps are switched on? So you automatically know, that we have two taps. We have less water coming in. So you will take longer to fill that pool. So it's an inverse proportion relationship or question. So I started by writing taps to time. So if three will take six hours, what did I do to get to two, I multiplied by two thirds, so I need to divide the six by two thirds. And if I divide that, it gives me nine hours. Remember to divide by a fraction, is equivalent to multiplying by the recipricals. Like saying six multiplied by three out of two. And it takes six hours to fill in the pool if only two types are used. And it makes sense we have less taps and it's taking more time. I could up here, also find out first, how long it would one tap, and then use that to have the answer for two taps. If you find it a bit tricky to use fractions, you could use that. Okay the other thing is that I know that they are inversely proportional, so the product should be the same. So three times six is eighteen. So two times something should also be eighteen. And therefore that something must be nine. I know that's wrong. Eighteen divided by two is equal to nine. So I've had the two answers now nine hours. Which is the third method, just to double check and to show you incase you have that. Started by saying what I'm going to say that, the taps is X and the time is Y. So the product of X, Y is eighteen. I'm going to substitute now, So two Y equals eighteen. Therefore, if I divide both sides of the equation by two, that will give me Y equal to nine. And in all three methods I had Y equal to nine. So did you get that as your answer as well, nine hours? Really good job. Well done. And now the third question. It usually takes Harry around seven hours to write two chapters of a novel. How many chapters would he expect to write in 24.5 hours? And what kind of question is this? He takes seven hours to write two chapters, more time in 24 and a half hours. Is he going to write more or less chapters. Of course he would write more chapters. So this here is a direct relationship, okay direct proportion relationship it's not inverse. As the number of hours increases, the number of chapters that he writes with also increase. So I can start by writing chapters to time. Two chapters take seven. So I'm going to find out one hour, how much is he going to write? He writes two sevenths. And then from that, use it to find 24.5, by multiplying by 24.5. And the answer is seven chapters. I did it using this method by finding first, how much he would write in one hour. You may have done it slightly differently or in a single step. And that is also correct. Let's have a look at our Explorer task. A farmer is building a rectangular pen for his pigs. The length and width are denoted by L and W. The area must be a hundred metres squared. Explain how you know that L and w are inversely proportional? If you're feeling confident to have a go at this question on your own, please pause the video now and have a go. If not, I'll be giving a hint in three in two and in one. My hint for you is to start by actually sketching what this would look like. So start by sketching a rectangle. And think about what could you use, as the length and width to make a hundred metres squared area. The easiest one is to have, a one by a hundred. Now try and use a different one and then a different one, and see what is happening to the length, and what's happening to the width. When the width is increasing, what's happening to the length? And are they increasing and decreasing by the same rate? With this hint, you should be able to make a start, please pause the video and have a go at the explore task. You need to spend 10 to 15 minutes on it. Resume the video once you have finished. There are so many things that you could have done with the explore task. I'm going to share with you my thoughts on this. So I started by saying, the area is equal to length multiply by the width. The question told us that the area must be a hundred. So really the product LW, is always going to be a hundred it's constant. And when it's constant we know that this means that LW are inversely proportional to one another. I then thought, okay, you know what, let me sketch. What would these rectangles look like? So first rectangle I can have is a one by a hundred, that would give me an area of a hundred. The second rectangle that I could have is two by fifty, that will also give me a hundred. And I quickly noticed here that, as I doubled one of the sides I have to the other. So when I doubled the length I have the width. Then I had to go at another one, and I had four by 25. And again I found out that, as I multiplied one of the sides by four, I divided the other side by four. And then I had five by twenty. Then I wrote it down as well just to double check. So one to a hundred, two to fifty, what have I done? I have doubled one I have the other. One to 100, four to twenty five. I multiplied by four, I divided by four. So I have done the inverse. Next one to a hundred, to move on to five to 20. I multiplied one side by five, I divided the other side by five. So I always did the inverse. So this show is that this is inversely proportional. Remember that you could have also used, non-integer numbers with this. It doesn't have to be only whole numbers. So this is how I've gone about with this question. I wonder how you do that? Okay. Well done for today's lesson, you've done some fantastic learning. Please remember to complete the exit quiz, to show what you know That's it from me for today. Enjoy the rest of your learning, and I'll see you next lesson. Bye.

It takes 4 builders 6 days to build a wall. How long will it take 2 builders?

It takes builders 6 days to build a wall. How long will it take 8 builders?

It takes 2 people 3 hours to clean the house. How long does it take 6 people to clean?

A journey takes 6 hours if driving at 30 mph. How long should the same journey if the driver drives ta 60mph?

True or False: The circumference of a circle and its diameter are inversely proportional?

Inverse proportion in context

In this lesson, we will interpret, solve and model solutions to inverse proportion problems.

Maths

Y8 Maths

Key Stage 3

Year 8

Direct and indirect proportion

Ratio and proportion

THEME

UNIT

Looking at proportional relationships

Direct proportion (Part 1)

Direct proportion (Part 2)

Direct proportion (Part 3)

Graphs of proportional relationships

The constant of proportionality

Proportion in context

Inverse proportion

Graphs of inverse proportion problems

Direct or inverse proportion?

Pens are sold in boxes. Each box contains 6 pens. How many pens are there in 5 boxes.

A chain costs 8p per 2cm. The cost and length are directly proportional. What is the cost of 6 cm?

Look at the table below. Choose the correct statement that describes the relationship between the number of mugs and the cost of the mugs.

The number of rulers are directly proportional to the cost. What is the missing number B?

x and y are directly proportional. What is the missing number in the box?

Which graph shows direct proportion?

Complete the sentence: When two quantities are in direction proportion, when one variable is zero, the other is .............

Which table shows direct proportion?

What is the constant of proportionality between m and n?

Decide whether the scenarios below represent direct or inverse proportion: The cost of apples and the quantity of apples.

Lesson overview:Inverse proportion in context

Core Content

Direct and indirect proportion: