Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

What an exciting day.

We're doing advanced problem solving with linear inequalities.

This sounds like an opportunity to show just how good we are, so let's get started.

Our learning outcome is that we'll be able to use our knowledge of inequalities to solve problems. We have some beautiful problems coming up for us today.

Key word we'll hear throughout the lesson, inequality.

And inequality is used to share that one expression may not be equal to another.

Two parts of the problems that we're solving today.

We're gonna begin by looking at what's called project constraints.

Graphing inequalities can help us solve complex-looking problems far more simply.

What do I mean by complex-looking problem? Well, I mean this one.

"Oak Academy is making two versions of a game.

The standard version takes 30 minutes to produce, the deluxe version takes 20 minutes to produce.

It takes 10 minutes to package the deluxe version and five minutes to package the standard version.

There are 18 hours available to make the games and four hours available to package them.

Because of market demands, we have to make at least twice as many standard games as deluxe ones.

What possible combinations of games could be made?" I don't expect you to answer that question right now, but by the end of this lesson, we'll be able to answer it really quickly.

Really simply.

This is the exact same problem.

We could begin by just considering some values.

For example, could we make 20 standard games and 10 deluxe ones? There's various constraints in this problem, one of which is production.

The standard version takes 30 minutes to produce, the deluxe version, 20 minutes to produce.

There's 18 hours available to make the games.

So, making 20 standard games and 10 deluxe ones, well that will be 20 lots of 30 minutes and 10 lots of 20 minutes.

That's 800 minutes.

It'll take us 800 minutes to make that many games.

We have 18 hours available.

That's 1,080 minutes.

So you can see now we can indeed produce this many games.

There's another constraint in packaging.

Takes 10 minutes to package the deluxe version, five minutes to package the standard version.

There's four hours available to package them.

20 lots of 5, and 10 lots of 10, that's 200 minutes.

Have we got 200 minutes? Absolutely.

We've got 240 minutes, so we can absolutely package these games.

One final constraint.

Because of market demand, we have to make at least twice as many standard games as deluxe ones.

Well, 20 standard games is twice as many as the 10 deluxe games, so we've satisfied that constraint.

So yes, we can make 20 standard games and 10 deluxe ones, but there's lots of spare time left unused.

It will be nice to know what greater volumes we could produce without having to do that amount of mathematics every single time.

I wonder how we'll manage this.

For now, let's just focus on production and packaging.

If we graph inequalities to represent our production and packaging potential, we could do this mathematics much quicker.

If we use x to represent standard games and y for deluxe ones, how might we express how long production takes? I'd like you to pause, do a little bit of thinking.

How might we express the length of time for production? Welcome back.

Hopefully you came up with the expression 30x + 20y.

That's x, the number of standard games, multiplied by 30, plus y, the number of deluxe games, multiplied by 20.

30x + 20y will give us the number of minutes it takes to produce x standard games and y deluxe games.

What's the limit for how long production can take? Well, that's 18 hours, but we want it in minutes, because our production time was calculated in minutes, so it's 1,080 minutes.

Therefore, what inequality will we graph to show the region of production constraint? Pause and see if you can predict the inequality we'll use here.

Welcome back.

Hopefully you said 30x + 20y is less than or equal to 1,080.

30x + 20y being the amount of minutes it'll take us to produce that many games, and that cannot exceed 1,080 minutes.

Quick check that you can do a similar skill to the one I just showed you.

What inequality will we graph to show the region of packaging constraint? Pause and come up with an inequality now.

Welcome back.

Hopefully you looked at these key bits of information from the problem.

10 minutes to package the deluxe version, five minutes to package the standard version, four hours available to package them.

Therefore, 5x + 10y is an expression for how many minutes it will take to package x and y games, and it cannot exceed 240 minutes.

So we need the inequality 5x + 10y is less than or equal to 240.

We can now graph our inequality for production constraint.

This is 30x + 20y is less than or equal to 1,080.

That graph will look like so.

We can also graph the inequality for packaging constraint.

I'm gonna put it on the same grid.

You'll see why.

There's our inequality for packaging constraint.

We can now quickly answer questions about capacity.

For example, can we make 10 standard games and 30 deluxe ones? Well, the point 10, 30 would be there on the graph, and you can see that we could produce them but not package them.

That point satisfies the production inequality, but does not satisfy the packaging one.

Could we make 35 standard games and 5 deluxe ones? 35, 5 would be there, and you can see that we could package them but not produce them.

That point does not satisfy the production constraint inequality.

Can we make 15 standard games and 15 deluxe ones? Absolutely, we could.

That point satisfies both inequalities.

We could produce 15 standard and 15 deluxe, and also package them.

This point is in the overlapping region where both inequalities are satisfied.

It is only values in this region which can be made, values in that region.

Quick check you've got this.

I'd like you to write statements about each of these points.

Are they within our project constraints? Three points, I'd like three statements from you.

Pause, and write them now.

Welcome back.

Hopefully of the point 40, 20 you said it satisfies neither inequality.

We can neither produce nor package that many games.

For 26, 13, I hope you said it satisfies only one inequality.

We can produce them, but not package them.

And for 30, 5, I hope you said it satisfies both inequalities, we can produce and package them.

Well done.

There's one more project constraint, which we have not yet graphed.

Let me bring the problem back up for you.

Can you spot it? What's the other project constraint that we didn't have on the graph that we just saw? Pause, see if you can spot it.

Welcome back.

Hopefully you spotted it's the last line.

Because of market demands, we have to make at least twice as many standard games as deluxe ones.

We somehow need to feature that on our graph of inequalities.

So the question I have for you now is what inequality do we need to graph? How are we going to graph that project constraint? Pause, see if you can come up with the right inequality.

Welcome back.

Did you come up with this? 2y is less than or equal to x.

If you think about that, y is the number of deluxe games.

We'd need to double it to get the number of standard games, but the number of standard games can be more than double the number of deluxe games.

That's why we end up with that inequality.

We can now add the inequality 2y is less than or equal to x to our graph.

When we do so, it looks like that.

We've got our production constraint, our packaging constraints, and our demand constraint graft on the same grid.

So we can quickly answer questions like this one.

Should we make 15 standard games and 10 deluxe ones? What do you think? Well done.

No, we shouldn't.

Whilst we could produce them, and we could package them, these numbers do not meet market demand, so we shouldn't be making those quantities.

Should we make 25 standard games and 8 deluxe ones? What do you think? Yes, this is an option.

We can produce them and package them, and these values of games meet market demand.

All three inequalities are satisfied by that point.

Quick check you've got this.

There's the same graph of the same problem.

My question for you is, which of these combinations can we, and should we, make? There's four to contemplate.

Pause and do so now.

Welcome back.

Hopefully you said no to option A.

Whilst we could produce that many, we couldn't package that many, and those values don't meet our market demand constraint.

For B, absolutely we could produce and package 25 standard and 10 deluxe.

Should we make them? Absolutely.

They meet our market demand constraint.

The same was true for C.

Whilst it might be unusual to make 36 standard games and no deluxe games, it's an option.

We can produce that many, package that many, and it still meets our market demand.

D was not an option.

We could package that many, produce that many, but those values don't meet our market demand.

Practise time now.

Question one.

It's a problem similar to the one that we've been working on throughout, but the numbers are different.

For part A, I'm gonna ask you to write inequalities for each project constraint.

For part B, you'll graph the inequalities that you wrote for part A, and shape the region which satisfies all project constraints.

For part C, you're going to use your graph to explain why we could make 30 standard games and 6 deluxe ones, but not 28 standard games and 10 deluxe ones.

Pause and do all of these things now.

Feedback time.

Let's see how we did.

Hopefully you used x to represent standard games and y to represent deluxe games, and came up with an inequality for the production constraint.

15x + 10y is less than or equal to 540.

And for the packaging constraint, 6x + 8y is less than or equal to 240.

and then on demand y is less than or equal to 12.

For part B, we were graphing those inequalities.

They would've looked like that on your graph.

For part C, we're using that graph to explain why we could make 30 standard games and 6 deluxe ones, but not 28 standard and 10 deluxe.

You might have said the point 28, 10 satisfies the production and demand inequalities, but not the packaging one, whereas the point 30, 6 satisfies all three inequalities.

On to the second half of our lesson now.

We're gonna be working with similar problems, but seeing what it means to be being optimal.

Another purpose of this mathematics is that it allows us to spot points where we are being optimal.

That means we are minimising wastage.

This is the same Oak Academy game scenario we saw earlier.

The shaded region satisfies the production, packaging, and market demand constraints.

A key question then is what is the maximum number of games we can make? Is it 25 and 5? Clearly not.

It's clearly not 25 standard and 5 deluxe games, because we have capacity from here to make more of each.

I can move that x value in a positive x direction, i.

e make more standard games, and I can move that y value in a positive y direction, i.

e.

make more deluxe games.

So if it's not 25 and 5, what is it? It's one of these points.

But my question for you to contemplate for a moment is which of these points is it? Where is our maximum number of games? Pause and have a think about that now.

Welcome back.

Did you spot it? It was 30, 9.

30 standard games, 9 deluxe games.

What's special about the point 30, 9? That's a key question, so let's explore it.

The edges of our region distant from the origin give us our highest production numbers.

The point 30, 9 is the intersection of our production and packaging inequalities.

It is this point which gives us our greatest number of games.

Let's compare these two points for a moment.

The point 24, 12 and the point 30, 9.

When you substitute the values of these two coordinate pairs into the inequalities, you understand what is so special about the point 30, 9.

Let's start by substituting 24, 12, x equals 24, y equals 12.

When those values go into our production constraint inequality, 960 is indeed less than or equal to 1,080.

It satisfies the inequality.

When we substitute those same values into the packaging inequality, we see something interesting.

240 is equal to 240, so it satisfies the inequality, but only just.

Can you see what's happened? We've not optimised all of the minutes available to us on production, but we have optimised all of the minutes available on packaging.

There were 240 minutes available for packaging.

We used all 240.

However, of the 1,080 minutes available on production, we didn't use them all.

We still got 120 minutes left.

That's two hours.

Could we use that time? I think you know the answer to that question already.

The point 30, 9, when substituted, does something lovely.

X equals 30, y equals 9.

When substituted into our production inequality, 1,080 is equal to 1,080.

It satisfies the inequality.

And when substitutes into the packaging inequality, we see the same thing we saw in the previous pair, 24, 12, every minute is utilised.

So, it's only at the point 30, 9 that we've used every available minute of production and packaging time.

That is why this is our optimal point, i.

e.

the moment when we produce the most games.

Quick check you've got this.

I'd like you to complete these statements about the coordinate pair 30, 9.

There's three key words missing from those statements.

Pause, see if you can fill them in.

Welcome back.

I hope you said this intersection is the moment where we're making the greatest number of games we can.

At this moment, we are using every available minute of production and packaging time.

This is our optimal point.

We've optimised all the capacity we have.

In the realm of being optimal, it's not just the number of games we make that we are seeking to optimise.

We're not making these games for the fun of making them.

We want to make a profit.

Let me give you this scenario.

If we make four pounds profit from the sale of every standard version, that's the x value, and 10 pounds profit from the sale of every deluxe one, that's the y value, is 30, 9 still our optimal point? What do you think? Can you work out the profit we make from each of these three points on the graph, and then tell me are we still being optimal at 30 and 9? Pause, have a think about this.

Welcome back.

Let's run through these figures.

When we make 36 standard and 0 deluxe, we make 36 lots of 4 pound profit, and 0 lots of 10 pound profit.

That a grand profit of 144.

When we make 30 standard games and 9 deluxe ones, that's 30 lots of 4 and 9 lots of 10, that's 210 pounds of profit.

How about 24 standard and 12 deluxe? It's fewer games, but it's 24 lots of 4 pound and 12 lots of 10 pound, which is 216 pounds.

From a profit perspective, our optimal point's different.

24, 12 is our optimal point from profit perspective.

Whilst we make fewer games, we have made more of the most profitable game, the deluxe one, so our overall profit is greater.

We need to use substitution to check the profit of each intersection, to find our most profitable point, or the point at which profit is optimised.

Quick true or false.

The point where we make the most products is the same point where we make the most profit.

Is that statement true or is it false? Once you've decided, I'd like you to use one of the two statements at the bottom of the page to justify your answer.

Pause, have a think about this now.

Welcome back.

I hope you said false, and justified that with not necessarily.

There may be a point where we've made fewer games, but can make more profit, because we've produced more of the most profitable game.

When we're being optimal, we have to look out for profit optimization, and not just production optimization.

Practise time now.

Question one.

This is the same project we modelled in task A.

You've seen this graph before.

For part A, I'd like you to answer what is the maximum number of games that can be made, and for part B, I'd like you to substitute into the inequalities to show why this is an optimal point.

Pause and do this now.

For part C, let me give you this scenario.

Three pounds profit is made in every standard game, the x value, and five pounds profit from the sale of every deluxe game, the y value.

What is the optimal point from a profit perspective? Pause, and see if you can work this one out.

Feedback time now.

Let's see how we got on with question one, part A.

What's the maximum number of games that can be made? That'll be that intersection there, 32, 6.

38 is the maximum number of games that can be made.

We find it at this point of intersection 32, 6.

For part B, I asked you to substitute into the inequalities to show why this is an optimal point.

When we substitute, x equals 32, y equals 6, into our production inequality, we see that 540 is equal to 540.

It satisfies the inequality only just.

Same for packaging.

240 is equals to 240.

That's the maximum value we can have to still satisfy that inequality.

We've optimised all of the minutes available to us for production and packaging when we make 32 standard games and 6 deluxe ones.

For part C, I asked you what is the optimal point from a profit perspective.

Was it 32, 6, the maximum number of games we could make, the point at which we are optimised on production and packaging? No, 32, 6 is not our optimal point from a profit perspective.

We make a lot of money, 126 pounds, but there's a point which beats it.

Hopefully you spotted 24, 12 is the optimal point of profit.

24 standard games and 12 deluxe games, that is 132 pounds of profit.

Once we make fewer games, we make a greater profit, because we made more of our more profitable game.

That's the end of the lesson now, sadly.

What have we learned? We've learned that we can use our knowledge of inequalities to solve problems in context.

For example, a project constraint problem.

We appreciate that intersections are important points, because they maximise our output from both a productivity perspective and or a profit perspective.

I hope you've enjoyed seeing how this topic in mathematics applies to this real-life situation, and I hope you look forward to using this maths in whatever future career you pursue.

I look forward to seeing you again very soon for more mathematics.