Lesson video

Hi everyone.

My name is Ms Ku and I'm really happy that you're joining me today.

Today we're going to look at direct and inverse proportion and lots of beautiful and interesting ways in which we show direct and inverse proportion.

I hope you enjoy the lesson, let's make a start.

Hi everyone and welcome to this lesson on problem solving with direct and inverse proportion under the unit: direct and inverse proportion.

And by the end of the lesson you'll be able to use your knowledge of direct and inverse proportion to solve problems. Let's have a look at some keywords.

Two variables are inversely proportional if there is a constant multiplicative relationship between one variable and the reciprocal of the other.

And two variables are in direct proportion if they have a constant multiplicative relationship.

Today's lesson will be broken into three parts.

We'll look at proportion with more than two variables, then other multistep proportion problems and then more complex matching problems. So let's make a start looking at proportion with more than two variables.

Now Laura says, "If a fuzzy is a buzzy, and a buzzy is a muzzy, is a fuzzy a muzzy?" These funny little riddles are classic examples of how variables can linked and one relationship with one variable can be linked with another relationship with another variable.

For example, if A is directly proportional to B and B is directly proportional to C, then we know A is directly proportional to C.

Alternatively, if A is directly proportional to B and B is inversely proportional to C, that means we know A is inversely proportional to C.

So now what I want you to do is complete the rest of these sentences.

If A is inversely proportional to B and B is directly proportional to C, that means we know A is, I want you to fill this in.

Next, if A is inversely proportional to B and B is inversely proportional to C, then we know A is, and I want you to fill that in.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well that means A should be inversely proportional to C, and the next question states that A should be directly proportional to C.

Really well done if you've got this.

So let's move on to some real life data.

Here we have spreadsheets showing some proportions and it shows the number of patients treated by a number of nurses.

It also shows the total vaccines given if each patient received two vaccines, and it also shows the total number of reported infections.

All are shown in our spreadsheet.

So let's start by expressing P in terms of N.

Firstly, we can see P and N are directly proportional because as N increases, P is increasing proportionally.

So I'm going to represent it here as P is directly proportional to N.

Given that P is directly proportional to N, that means I can write it in the form of P is equal to KN.

Now we can substitute any value for N with any corresponding value for P.

I've just chosen N to be three and P to be 96.

That means I can work out K to be 32, thus giving me the equation of P in terms of N to be P is equal to 32N.

Now we can also see this multiplier 32 using our spreadsheet.

You can see it here.

Now let's have a look at V and P and we're asked to express V in terms of P.

We know each patient received two vaccines.

Now given this, that means we know the equation of V in terms of P must be V is equal to 2P.

Alternatively, we can use the table again and you can spot if we multiply P by two, we get the number of vaccines.

So now we know V is proportional to P and P is proportional to N.

So that means we know V is proportional to N.

That means we can express V in terms of N.

So given the fact that we know V in terms of P is V equals 2P, and we know P in terms of N is P is equal to 32N, that means we can write V in terms of N by substituting P as this is given in terms of N.

So knowing V is equal to 2P that substitute the P as 32N, which I've done here.

So that means I know V is equal to two, multiply by 32N.

So now I have V in terms of N, which is V is equal to 64N.

Continuing with this, we can also find the proportional relationship between R and V.

I want you to have a look at our spreadsheet and tell me do you think it's direct or inverse proportion? Well, you can see R and V are inversely proportional because as V increases, R is decreasing proportionally.

So let's express R in terms of V.

Given the fact that we know R is inversely proportional to V, I can express it as an equation.

R is equal to K over V.

I can substitute any value I want for R as long as it's with the corresponding value of V.

All I've done is chosen R to be 250 and V to be 192.

Now substituting this in, I can work out my K value and my K value is 48,000.

So that means I know R in terms of V is R is equal to 48,000 over V.

Now given that we know R is inversely proportional to V and V is directly proportional to N, that means we know R is inversely proportional to N.

So we can express R in terms of N.

Remember R is equal to 48,000 over V and we know V is equal to 64N.

So we can write R in terms of N by substituting V as this is given in terms of N.

R is equal to 48,000 over V, we know V is 64N.

So that means I've simply substituted my V as 64N into my equation, thus giving me R is equal to 750 over N.

Well done.

So let's have a look and a check.

We know A is directly proportional to B and we know B is inversely proportional to C.

What I want you to do is write an equation for A in terms of C, given that A is equal to 5B and B is equal to 12 over 25C.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well we know A is equal to 5B, so we're gonna substitute B into A.

So that means we know A is equal to five multiplied by our 12 over 25C, which gives us A is equal to 12 over 5C.

We have now written A in terms of C.

Let's have a look at another check.

Here we know Y is proportional to P.

When Y is equal to 24, P is equal to eight.

So part A wants you to write Y in terms of P.

Now we're also told that P is directly proportional to X and when P is equal to seven, X is equal to 1.

4.

Part C wants you to write Y in terms of X.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, we know that Y is directly proportional to P, so it should be Y is equal to KP.

Substituting our Y and P value, we can work out K to be three so we now have our equation, Y is equal to 3P.

For part B, we know P is directly proportional to X, so that means P is equal to KX.

Substituting what we know we can work out K to be five, giving me the equation P is equal to 5X.

Now given the fact that we know Y is equal to 3P and we know P is written in terms of X, we can substitute that P for 5X giving me Y is equal to three multiply by or 5X, giving me Y is equal to 15X.

Really well done if you've got this.

And now I just want to take a minute just to point something out.

Notice how we used a lower case k for part A and then an upper case K for part B.

Now this wasn't essential, but given the fact that we're looking at two different constants, using a different constant is good practise as k and capital K are two different values.

So connecting two or more proportional relationships can also involve complicated roots or exponents.

And this process still remains the same, you just need to simply build on those inverse operations.

For example, A is directly proportional to B squared and B is inversely proportional to the square root of C.

Now when A equals 100 and B equals five, when B equals 12, C equals 64 and the question wants us to write A in terms of C.

Just like before we're going to start by writing the equation for A in terms of B.

We know this is a directly proportional relationship, so we can write it as A is equal to K multiply by B squared, substituting what we know, we can work out K to be four, thus giving us the equation for A in terms of B, A is equal to 4B squared.

Now for the second part, B is 12 and C is 64 and we're told this is inversely proportional.

So therefore B is equal to K over the square root of C, substituting in what we know and working out the value of K, we can work out K to be 96.

So we now have our equation for B in terms of C, B is equal to 96 over the square root of C.

From here we can write A in terms of C.

We know A is equal to 4B squared, if B is equal to 96 over the square root of C, I'm simply substituting B in terms of C into our equation for A.

So we have four multiply by 96 over the square root of C all squared.

Working this out using the priority of operations, we know A is equal to four multiplied by 9,216 over C, which simplifies to give A is equal to 36,864 over C.

So now we have A in terms of C.

Now it's time for a check.

Izzy was given this question, X is proportional to Y and Y is proportional to Z.

She's told that when X is 20, Y is 120 and when Y is 16, Z is 3.

2 and she's asked to write Z in terms of X.

Here's her working out.

I want you to spot her mistake and I also want you to correct the answer, see if you can give it a go.

Press pause for more time.

Great work.

Let's see how you got On.

Well hopefully you've spotted the question wants Z in terms of X.

She's written X in terms of Z, therefore the correct equation is Z is equal to six fifths X.

Really well done if you got this.

Great work everybody.

Now it's time for your task.

Take your time and work out the answers to question one and two.

Press pause if you need more time.

Well done.

Let's move on to question three and four.

Same again, take your time, read the questions carefully, press pause if you need.

Well done.

Let's have a look at question five.

Place a tick in the table to identify if the following statements are true or false.

Press pause as you'll need more time.

Well done.

Let's go through these answers.

Here's my working out and here are the answers.

Press pause if you need.

Great work.

Question three and four, here are the answers, press pause if you need more time to mark.

Well done.

And for question five, you should have had these ticks.

Really well done.

Great work everybody.

So now let's move on to other multi-step proportion problems. When recognising direct or inverse proportions, we must also consider the impact on variables given proportional changes.

For example, there is a directly proportional relationship between X and the square root of Y.

If we were to increase Y by 69%, the question asks us to work out the percentage increase of X.

Well, the original equation shows X is equal to K the square root of Y.

This equation can be formed because we know X is directly proportional to the square root of Y.

Now don't worry about the value of K, we do not need to work it out as we only need to find the impact of the proportional change of Y.

Now given the question states we're increasing Y, by 69% an increase of 69% means we have a multiplier of 1.

69.

So that means we multiply our Y value by 1.

69.

Now using our knowledge on surds, in other words the square root of AB is exactly the same as root A multiplied by root B.

We can write the square root of 1.

69Y as the square root of 1.

69 multiplied by the square root of Y.

Then we can work out the square root of 1.

69.

This is 1.

3.

Now you can clearly see when Y has been increased by 69%, we have a multiplier of 1.

3.

This means when Y has increased by 69%, X increases by 30%.

Now let's have a look at a check.

There is a directly proportional relationship between P and the cube root of T.

If T were to increase by 33.

1% we're asked to work out the percentage increase of P.

See if you can give it a go Press pause if you need more time.

Well done.

Let's see how you got on.

Well the original equation shows that P is directly proportional to the cube root of T, so therefore P is equal to K multiplied by the cube root of T.

Once again, we don't have to worry about working out the value of K, let's identify the increase of 33.

1%.

Well T has been increased by 33.

1%, so that means T has a multiplier of 1.

331.

Remember our rules associated with surds, we can rewrite this as the cube root of 1.

331T as the cube root of 1.

331 multiplied by the cube root of T.

Then we can work out the cube root of 1.

331.

This is 1.

1.

So we can clearly see from our new equation that when T increases by 33.

1%, P has increased by 10%.

Well done if you've got this.

Remember, proportional changes can also include decreasing a variable.

However it is important to ensure you use the correct multipliers.

For example, two Oak pupils have made a common error here.

Can you spot them? The question states that Y is proportional to X squared.

And when X is reduced by 30%, the question asks, what's the overall percentage decrease of Y.

Alex has done the top working out and Jun has done the bottom working out.

Press pause if you need more time to have a look at where their common error lies.

Well done.

Let's have a look at Alex's mistake.

While the decrease of 30% means we have a multiplier of 0.

7, not 0.

3.

Let's have a look at Jun's mistake.

Well if 0.

49 is the multiplier, then that means Y has been decreased by 51%.

So if we were to show the correct working out, a decrease of 30% is a multiplier of 0.

7.

Then working this out, you can see Y is equal to K multiply by a 0.

7X all squared.

So we have Y is equal to 0.

49KX squared, meaning Y has been decreased by 51%.

Really good if you got this.

Let's have a look at another check.

I want you to fill in the blanks where we know the original equation is Y equals KX squared.

We're doing something to X by a certain percentage and it gives you Y is equal to K multiply by 1.

2X squared, which then simplifies to Y is equal to 1.

44KX squared.

This means Y is being what by what percentage? Let's have a look at the second question.

The original equation is Y equals K multiplied by the square root of X.

What are we doing to X and by what percentage, given this working out? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, we should have got, X is increased by 20%.

Doing this working out you can see that this means Y has been increased by 44%.

The second question states that that X has been decreased by 43.

75%, which means Y has been decreased by 25%.

Massive well done if you've got this.

Great work everybody.

So now it's time for your task.

I want you to work out the percentage increase or decrease of Y and shade this number in to reveal a lovely little word in our grid.

Take your time and press pause when you are ready.

Fantastic work everybody.

Let's have a look at those answers.

Well hopefully you've got these answers and you've shaded it to reveal the wonderful word pie.

Well done.

Great work everybody.

So let's move on to more complex matching problems. In order to match graphs and equations, it's important to remember the key features of different types of graphs.

Before looking at these features, let's recap on some keywords associated with linear graphs, quadratic graphs, cubic graphs, exponential graphs and trigonometric graphs.

Let's recap on the word turning points.

Now the turning point of a graph is a point on the curve where, as X increases the Y value changes from decreasing to increasing or vice versa.

So what I want you to do is identify how many turning points to these graphs have.

Press pause as you'll need more time.

Well done.

Well the first one has one turning point, it is a parabola, well done.

The second one has no turning points, it's a straight line.

The third one is a cubic graph and it has two turning points.

The fourth is a reciprocal graph and it has no turning points.

Really well done if you spotted this.

This is a sine graph and it's important to note sine and cosine have an infinite number of turning points, but in this graph we can only see two.

Really well done if you got this.

Now what I want you to do for question A sketch two different graphs with exactly one turning point.

Bonus points if you can state the equation of the graph.

For B, sketch two different graphs with exactly two turning points.

Once again, bonus points if you can state the equation of the graph.

And for C, sketch three different graphs with no turning points.

See if you can give it a go.

Press pause if you need more time.

Well done.

Here are just some examples of graphs with one turning point, two turning points and no turning points.

Very well done if you've got any of these, especially well done if you've given the names of those equations.

Now let's have a look at another check.

When Y is directly proportional to X, the graph of the relationship has no turning points.

Do you think this is true or false? And to justify your answer, do you think it's A, graphs showing variables in direct proportion can be nonlinear and have turning points? Or do you think the reason is because graphs showing variables in direct proportion must be linear and intersect at 0,0? What do you think? Press pause if you need more time.

Well, it is true.

When Y is directly proportional to X, it has no turning points, and this is because that relationship is shown as linear, intersecting 0,0.

Really well done if you've got this.

Another key piece of information to remember when matching graphs with their equations are intersections on the axis.

For example, what do you notice about all the coordinates of our graph when it's crossed the X axis? Have a little think.

Well, when the graph touches or passes through the X axis this means Y is equal to zero.

So when a graph passes through or touches the X axis, the X coordinate is also called a root.

From this sketch you can see the roots are, X is equal to negative two, X is equal to one and X is equal to three.

And we can use these roots to identify the equation of the graph in factorised form, X plus two, multiply by X minus one, multiply by X minus three.

Now Aisha says, "I know these are both cubic, but how do I know which equation matches with which graph?" So let's see if we can help.

We know the intercept on the X axis means Y is equal to zero.

So we can find the roots or the intercepts on the X axis by substituting Y is equal to zero.

And then solving for X.

So let's have a look at Y is equal X cubed subtract 4X.

Making Y is equal to zero, we have zero is equal to X cubed subtract 4X, then it can factorised.

This means zero is equal X multiply by X squared takeaway four.

I spot a difference to two squares, so that means I have X multiply by X minus two, multiply by X plus two.

In other words, I have three roots, three interceptions on our X axis.

X is equal to zero, X is equal to two and x is equal to negative two.

Now let's have a look at Y equals X cubed, equating to zero to find those interceptions on the X axis.

Well that means my only root would be X equals zero.

So that means I can pair up my graph with my equation.

Aisha says, "I can now match them," because she knows the roots, she knows them because she set Y to be equal to zero.

Moving on, what do you notice about all the coordinates of the graph when they cross the Y axis? And how can you quickly see which graph shows a directly proportional relationship? Have a look at these graphs and have a little think.

Well, when a graph passes through or touches the Y axis, this means X is equal to zero.

So when graphs show a direct proportion that always an intersection at 0,0.

Now these do not show direct proportion.

They don't show direct proportion because there is no intersection at 0,0.

And it's also important to note.

for direct proportion as Y increases, the X in the end must also increase proportionally.

Now let's have a look at a check.

When X is greater than zero, I want you to match the type of graph and identify if they show a proportional relationship.

And if they do not show proportionality, I want you to explain Y.

See if can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, five, multiply by the square root of X is this graph here.

We have Y is equal to 7X squared is given here, Y is equal to two over X is given here, Y is equal to sin X is given here and lastly Y is equal to 3X subtract four is given here.

The first one does show proportional relationship as does the second one.

The third one does not show proportionality.

And this is because as the X and Y are not in proportion, as the X increases, the Y is increasing and decreasing.

The fourth graph does not show direct proportion, it shows inverse proportion.

And lastly, the last graph does not intersect at 0,0, so it does not show direct proportion.

Amazing work everybody.

So now let's have a look at your task.

I want you to match the description to the graph in the table.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question two.

Write down the letter of the graph which has these equations.

So if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question three.

Take your time, read each of these questions carefully.

You get a bonus mark if you can state the equation of the graph as well.

Press pause as you'll need more time.

Great work.

Let's have a look at these answers.

Here are our answers to question one.

Press pause to mark.

For question two, here are our answers for question two.

Great work.

Press pause to mark.

For question three, here are some example sketches.

Massive well done if you got this one right.

C was particularly difficult.

Great work everybody.

So in summary, we can connect two or more proportional relationships by forming the equation of proportionality and substituting one equation into another.

We have also looked at changing the proportion of one variable and working out how it changes the proportion of the other variable.

Lastly, we've also looked at matching more complex graphs that show proportional and nonproportional relationships.

Great work everybody.

It was wonderful learning with you.