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 finding the constant of proportionality for inverse proportion, under the unit direct and inverse proportion.

And by the end of the lesson you'll be able to use the general form for an inversely proportional relationship to find k.

We'll be looking at some keywords starting with inversely proportional.

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

Today's lesson will be broken into two parts.

We'll be expressing proportionality, algebraically, and then moving on to solving inverse proportion problems. So let's start by expressing proportionality algebraically.

Now notation in mathematics is very important.

We use symbols and notation because they're easier to read and understand, they're concise and take up less space, they can be used to represent complex concepts and it allows mathematical ideas to be communicated more effectively than words.

So when we use this sign to show proportion, this symbol means there is a proportional relationship between the variables.

For example, this means y is proportional to x.

This means a is proportional to b.

We know two variables are inversely proportional when there is a constant multiplicative relationship between one variable and the reciprocal of the other.

And we can represent this relationship using this proportionality symbol.

For example, y is equal to seven x.

This means y is proportional to the reciprocal of x, but we say it as y is inversely proportional to x.

Let's have a look at another example.

w is equal to 5 over 9p.

This means w is proportional to the reciprocal of p, but we say it as w is inversely proportional to p.

Now it can also be written as x is inversely proportional to y or p is inversely proportional to w.

Now Laura says, what about the seven and the five ninths? Well these are the constant, sometimes called the constant of proportionality and they are important but they remain constant as the variable increases or decrease.

Therefore, we only compare the relationship between the variables, not the constants.

For example, here we have P is equal to four over 7q.

This means p is inversely proportional to q.

This means four sevenths is the constant of proportionality.

So once we identify variables are directly or inversely proportional, we know there will always be a constant of proportionality.

Therefore, if y is inversely proportional to x, then we can rewrite it as an equation, y is equal to k over x where k is the constant of proportionality.

What I want you to do is do a quick check question.

I want you to fill in given some information about the equation or and the notation or and the constant k, I've done the first one for you, y is equal to five over p, this means y is inversely proportional to p and we have a constant of proportionality to be five.

See if you can give the others a go.

Press pause if you need.

Well done.

Let's have a look at b.

Well m equals four over 7j.

This means m is inversely proportional to j and we have the constant k to be four sevens.

Next we know that t is inversely proportional to v and we know we have a constant of proportionality to be 12 over 11, so therefore the equation must be t is equal to 12 over 11v.

Next we have p is equal to the square to three over 2r.

This means we have p is inversely proportional to r and the constant of proportionality is root three over two.

Lastly, great question, well done, have you got this? We have the equation, z is equal to pi over t squared.

We know z is inversely proportional to t squared and the constant of proportionality is pi.

Well done if you got this.

When we are given variables which are inversely proportional, we are able to work out the constant of proportionality.

For example, p is inversely proportional to q.

When p is equal to two and q is equal to 0.

4 we're asked to express p in terms of q and then we're asked to work out p when q is 12 and then we're asked to work out q when p is equal to one.

So first of all, here we are told that p is inversely proportional to q, so therefore we know we can rewrite this inversely proportional relationship as this equation.

Now we can work out k by simply substituting our values of p and q.

This means my equation was originally p is equal to k over q and now substituting those values I have two is equal to k over 0.

4.

From here we can solve for k.

This means k is equal to two multiplied by my 0.

4.

So now I have k is four fifths.

So we have an expression of p in terms of q, p is equal to four over five q.

Now for part b, we simply substitute it into the formed equation.

So we're trying to work our p when q is 12, substituting it into our equation, p is equal to four over five times 12, which gives me p to be four sixtieths, which then simplifies to p to be one 15th.

For part c, we simply substitute and solve for q.

So knowing p is equal to four over five q I know for part c, p is equal to one.

So I've substituted this in, forming an equation.

One is equal to four over 5q.

From here I'm solving for q, giving me q is equal to four fifths.

Forming the equation allows us to work out any value of one variable given the knowledge of the other variable.

Press pause if you'd like to copy this down.

Well done.

Let's move on to a check.

T is inversely proportional to rr, t is equal to 24 when r is equal to five.

Part a wants you to write an expression for t in terms of r.

Part b wants you to work out t when r is 20 and part c wants you to work out r when t is 50.

See if you can give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well first things first, we know given that t is inversely portion to r, we can write the equation in this form using our constant k.

T is equal to k over r.

Substituting our values, we have 24 equals k over five.

Working out the value of k to be 120.

So that means I know my expression for t in terms of r is t is equal to 120 over r.

From here I can simply substitute my r value for 20 and workout t.

So t is equal to 120 over r, so therefore t is 120 over 20 and that gives me the t value six.

For part c using the same formula, t is equal to 120 over r.

I'm substituting when t is equal to 15, rearranging to make r the subject, I have my solution for r to be eight.

Really well done if you've got this.

Great work everybody.

Now it's time for your task.

I want you to fill in the table, press pause if you need more time.

Well done.

For question two, I want you to work out the answers to these two questions given the fact that m is inversely proportional to p and for question three, y is inversely proportional to d.

So you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question four, x is inversely proportional to w.

Same again, write an expression for x in terms of w, work out x when w is equal to three and give your answer in surd form.

And for c, work out w when x is 10, same again, give your answer in surd form.

See if you can give it a go.

Press pause if you need more time.

Great work.

Let's do question five.

Put a tick in the box if you know it's always true, sometimes true or never true.

press pause if you need more time.

Well done.

Let's go through these answers.

You should have these answers.

Press pause to copy them down.

Great work.

Moving on to question two.

Here's my working out, press pause if you need more time to copy them down.

For question three, here's my answers, once again, press pause if you need more time to copy them down.

For question four, here are the answers.

Great work if you've got this one, the value of a is five and the value of b is two.

And for part c, the value of a is three over two and the value of b is two.

Really well done if you've got this.

Lastly for question five, this question helps you think deeply about that inverse proportional relationship between variables.

Really well done if you got this.

Great work everybody.

So now let's look at the second part of our lesson solving inverse proportion problems. Now what do you think the following mean? Have a look, press pause if need more time.

Well done.

Well for a it means y is proportional to the reciprocal of x squared, but we also say y is inversely proportional to x squared.

For b we say p is proportional to the reciprocal of the cube root of q or more commonly said as p is inversely proportional to the cube root of q.

Lastly, t is proportional to the reciprocal of the square root of m cubed or more commonly we say as t is inversely proportional to the square root of m cubed.

Well done if you got this.

So the inverse proportional equations always take this form, y is equal to k over x to the power of n.

In the previous part of the lesson we looked at when n was equal to one, but now we're going to move on where n is equal to any value.

Now writing these equations in this form is generally a good approach when solving problems. Therefore working out the constant proportionality is important to find the value of a variable.

So let's have a look at an example.

Here we have y is inversely proportional to the squares of x and we're given the fact that x is 2.

25 when y is equal to 12 and we're asked to work out y when x is 0.

25 and we're asked to work out x when y is equal to nine.

So firstly we can rewrite this inversely proportional relationship as an equation y is equal to k over the square root of x.

Then we can work out k by substituting those known values for x and y.

So we know x is 2.

25 and y is 12.

So all I've done is substitute them in.

From here I've got 12 is equal to k over the square to 2.

25, which gives me 12 is equal to k over 1.

5.

Now I can solve for k giving me the value of k to be 18.

So now I have my expression for y in terms of x, y is equal to 18 over square root of x.

Now let's have a look at question a.

Question a wants us to work out the value of y when x is equal to 0.

25.

So I've simply substituted, so y is equal to 18 over the square root of 0.

25 which gives me y is equal to 18 over 0.

5, which then tells me y must be 36.

For part b, same again, we're still using the same equation.

y is equal to 18 over the square root of x, but now we can substitute when y is equal to nine.

So we have nine is equal to 18 over the square root of x.

Rearranging to make the x term the subject we have the square root of x is equal to 18 over nine.

So therefore the square root of x is equal to two.

So that means x is equal to two squared giving me x to be four.

Great question, copy it down if you need.

Now it's time for your check, given that h is inversely proportional to y squared when y is equal to five, h is equal to 0.

96.

Part a wants you to work out h when y is equal to 20 and part b wants you to work out the positive value of y when h is equal to six.

See if you can give it a go.

Press pause if you need more time.

Well done, let's see how you got on, where we know h is inversely proportional to y squared so therefore h is equal to k over y squared.

From here we can substitute our values, h is 0.

96 and y is equal to five.

So we have 0.

96 is equal to k over five squared.

Working this out we can then solve for k where k is 24.

Now we have our equation h is equal to 24 over y squared.

Now for part a you need to work out h when y is 20.

So substituting when y is 20, we have h is equal to 24 over 20 squared, thus giving me h is equal to 0.

96.

Next, let's work out b.

For part b it says six is equal to 24 over y squared, y squared therefore must be 24 over six, therefore we have y squared is equal to four and the square root of four is two.

Well done if you got this one right.

Therefore we know inverse proportion relationships in this form, y is equal to k over x to the n allow us to solve problems involving inverse proportion and it's important to questions can be represented in different ways, but once the direct equation or inverse equation of proportionality has been identified, unknown variables can be easily found.

So let's have a look at an example.

What do all these questions have in common? We have the hours h to complete a task are inversely proportional to the square number of robots y squared and it takes two robots six hours to complete the task.

How many hours will it take six robots? B says h is proportional to one over y squared and we have this table and c, it says h is proportional to one over y squared or y equals five, h equals no 0.

96.

Have a little think.

What do all these questions have in common? Well done.

Well, hopefully you spotted they all show h is inversely proportional to y squared.

They also show the same equation, well done if you work this one out, they show that h is equal to 24 over y squared.

So knowing that all these are variations of h is equal to 24 over y squared, let's work out these answers.

The first part says the hours, h, to complete a task are inversely proportional to the square number of robots y squared.

It takes two robots six hours to complete the task.

How many hours will it take six robots? Well we know we have the same formula, h is equal to 24 over y squared.

So substituting y is six.

We have h is equal to 24 over six squared.

That means number of hours is 24 over 36, so it will take two thirds of an hour.

For b, we have this table showing h is inversely proportional to y squared.

So recognising this, we can substitute into our formula, knowing that h is equal to 24 over y squared, I know h is 150.

So substituting this into our equation, I can rearrange to make y the subject of the, giving me y squared is 24 over 150, working out y to be 0.

4.

From here we can insert into our table.

Mathematically y could have also been negative, 0.

4, but the context is ours.

So we ignore the negative solution here.

Really well done if you spotted this.

And lastly, we can work out the value for h when y is equal to 10 by simply substituting it in and working h to be 0.

24.

Once again inserting into our table, we have our complete table.

Well done if you got this.

Great work everybody.

So now it's time for your task.

The time to complete a task is inversely proportional to the square root of the number of people involved in the task and it took 49 people to do the task in 50 hours.

How many people would be needed to do the task in 17.

5 hours? So you can give it a go.

Press pause if you need more time.

Great work.

Let's see how you got on.

Well we know t is inversely proportional to the square root p, rewriting it as an equation, we have t is equal to k over the square of p.

Now we can substitute t to be 50 and our people to be 49.

So we have 50 is equal to k over the square root of 49.

From here we have 50 is equal to k over seven and we can work out our constant of proportionality to be 350.

Now I have my equation t is equal to 350 over the square of p.

Now we can just substitute in the hours of 17.

5.

So that means we have 17.

5 is equal to 350 over the square root of p making the p term the subject we have the square root of p is 350 over 17.

5, which gives me the square root of p is equal to 20.

Therefore I know p is equal to 400.

So that means we know it would take 400 people to complete the task in 17.

5 hours.

Really, really well done if you've got this, let's have a look at another check.

Here we have a table showing that p is inversely proportional to q cubed.

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 we can rewrite it as an equation, p is equal to k over q cubed.

From here we can substitute some values, but when p is 24, I know q is 10.

So these are the only two values that I can substitute in.

Therefore 2.

4 is equal to k over 10 cubed.

I can work out k to be 2,400.

Now I know my equation to be p is equal to 2,400 over q cubed.

Now I can fill in the table using some values we have.

We know that q is 0.

2, so substituting it in, I can find p to be 300,000.

I know p is 19.

2, so substituting it in, I can work out q to be the cube root of 125, which is five.

Really well done if you've got this.

Great work everybody.

So now it's time for your task, given that p is inversely proportional to the cube root of m, work out p when m is eight and work out m when p is 200.

See if you can give it a go.

Press pause for more time.

Well done.

Let's have a look at question two.

We've got some tables here and some inversely proportional relationships.

Take your time, press pause if you need.

Well done.

Let's move on to question three.

Question three, match the correct equation with the table.

See if you can give it a go.

Press pause for more time.

Well done.

Let's move on to question four.

Great question, take your time, read it carefully.

Press pause as you most certainly need more time.

Well done.

Let's go through these answers.

Well, for question one, I've put this working out on the screen and we have an p value to be 20 for part a, 0.

008 for part b.

For question two, we should have had these values here and these equations.

Really well done if you got this.

Now for three, we have matched these equations with these tables.

And for d, this was a great question, giving t to be 0.

032 hours.

Massive well done if you got this.

Great work everybody.

So in summary, we use this proportionality symbol to show proportion, and this symbol means there is a proportional relationship between variables.

We know inverse proportion equations can be written in the form y equals k over x to the n, where k is known as the constant of proportionality.

For example, p is equal to four over seven, q to the four.

This means p is inversely proportional to q to the four and it also means four over seven is the constant of proportionality.

Lastly, finding the direct or inverse equation of proportionality allows you to calculate unknown variables easily.

Really, really well done today.

It was great learning with you.