Lesson video

Hello everyone, my name is Ms. Ku, and I'm really happy that you're joining me today because it's one of my favourites again, compound measures.

Compound measures is one of my favourites because it's the real life application of mathematics.

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

Hi everyone and welcome to today's lesson on combining speeds, and it's under the unit, compound measures.

And by the end of the lesson you'll be able to combine compound measures, calculating overall speed.

So let's have a look at our keyword, speed.

Now speed is the rate at which something is moving, and it's measured as a distance travelled per unit of time, and we'll be looking at this a lot during our lesson.

Today's lesson will be broken into two parts, combining speeds, and then solving problems involving speed.

So let's make a start, but before we start looking at combining speeds, I just want to recap on combining averages first.

For example, let's say we have a list A, which consists of the numbers two, three, four, five and five.

This means the mean of list A would be the sum of those numbers, 19, divided by how many numbers we have, which is five, thus giving me a mean of 3.

8.

Now let's have a look at list B.

List B consists of the numbers six, 10, 10, and 10.

To work out the mean of this, sum them to give 36, divided by all four values, giving me a mean of nine.

So how would you find the mean of both list A and list B? Have a little think.

Well, Andeep says, he would work out 3.

8 add nine, and then get that answer and divide it by two, giving me 6.

4.

But Sophia says, well she's gonna sum the values of A and the values of B.

So 19 add 36, which is 55, and then divide by the total numbers in both A and B.

So you can see we have nine values there, giving a mean of 6.

1 to one decimal place.

Which approach do you think is correct? Well done.

Hopefully you spotted, unfortunately, Andeep is incorrect.

Andeep's working out is incorrect because he's actually working out an average of the averages, and this is a really common error.

Sophia is correct as she's using the sum of each list and then correctly dividing by the number of data values.

Now let's have a look at how this links to combining speeds.

We've got Laura, who runs for 15 minutes at an average speed of four kilometres per hour.

She then runs for five minutes at an average speed of six kilometres per hour, and the question wants us to work out Laura's average speed.

Now Andeep once again says, "I would work out four add six, get that answer divided by two to give five kilometres per hour," but I want you to explain why is this incorrect.

Press pause if you need.

Well, hopefully you spotted, once again Andeep is working out the average of the average, that this is incorrect.

So let's work it out properly.

Well firstly, we need to work out the distances travelled.

So we know Laura travels at an average speed of four kilometres per hour.

So you can see I've inserted this into our ratio table, and we also know that one hour is 60 minutes, so all I'm gonna do is convert my one hour into 60 minutes.

We know from the question that Laura runs for 15 minutes, so we need to convert our 60 minutes into 15 minutes, and to do this, we simply divide by four.

This means that for those 15 minutes, Laura travelled a distance of one kilometre.

Now let's have a look at what she did for the second part.

Well, we know she travelled six kilometres per hour, so I've put this in our ratio table, six kilometres per hour.

Once again, we know 60 minutes is one hour, so simply converted our one hour into 60 minutes.

Remember, she runs for five minutes at that speed.

So to work out how far she's travelled in five minutes, we simply divide by 12, meaning that she's travelled 0.

5 kilometres in five minutes.

So that means Laura travelled one kilometre in 15 minutes and then she travelled 0.

5 kilometres in five minutes.

So that means we know Laura travelled 1.

5 kilometres in 20 minutes.

So I'm gonna put this in our ratio table.

The total distance was 1.

5 kilometres and the total time was 20 minutes.

But remember, speed is measured per unit of time, so I need to convert our 20 minutes into an hour.

To do it, I multiply by three, thus giving me 4.

5 kilometres per 60 minutes, which is 4.

5 kilometres per hour.

So that means Laura's average speed is 4.

5 kilometres per hour.

So summarising, when asked to find the average speed, it's important to know the total distance travelled and the total time taken, as this will allow you to calculate the average speed.

And ratio tables are such concise and neat ways to show all structured working out, but remember, they're not essential.

So let's have a look at a quick check.

A car travels from A to B in 10 minutes at a speed of 42 kilometres per hour.

Then the car travels from B to C in 40 minutes at a speed of 60 kilometres per hour, and you're asked to work out the average speed of the car in kilometres per hour from A to C.

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, let's look at the first part of the journey, A to B.

We know the speed was 42 kilometres per hour, in other words, 42 kilometres per 60 minutes.

Considering we wanna find out how far the car travelled in 10 minutes, we're going to simply divide by six, thus giving me, seven kilometres were travelled in 10 minutes.

Looking at B to C, we know the speed was 60 kilometres per hour, which is the same as 60 kilometres per 60 minutes.

To find out how far the car travelled in 40 minutes, I simply multiply by 2/3, thus giving me 40 kilometres per 40 minutes.

So that means the total distance from A to C is 47 kilometres, and the total time was 50 minutes.

In this question, the unit of speed is given as kilometres per hour, so let's find out this distance travelled in 60 minutes.

Well to do this, we multiply it by 6/5 to give me 56.

4 kilometres per 60 minutes, which then gives us the average speed from A to C to be 56.

4 kilometres per hour.

Really well done if you've got this.

Great work everybody, so now let's have a look at another check question.

The Oak teacher travels 220 miles in six hours and 40 minutes in his car.

He then cycles for 80 minutes at a speed of 15 miles per hour.

The question asks, what is the Oak teacher's average speed for the whole journey? 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, looking at the first part of our journey, we know the total distance was 220 miles and the total time was 400 minutes.

Notice how I've converted them all to minutes.

So this was quite nice, as we didn't have to do much converting here as we have our distance and time.

For the second part of our journey, we know the speed was 15 miles per hour, so this is 15 miles per 60 minutes.

Well, we need to work out the distance for 80 minutes.

To do this, we multiply by 4/3, giving me a distance of 20 miles in our 80 minutes.

That means we know the total distance is 240 miles and the total time is 480 minutes.

And remember, looking at our units of speed, it's miles per hour, so we need to convert it into hours.

To do this, we're going to convert our 480 minutes into 60 minutes, so we're dividing by eight, thus giving me 30 miles per 60 minutes, so now we have our average speed, is 30 miles per hour.

Really well done if you've got this.

Great work everybody, so now it's time for your task.

I want you to fill in the table to give the average speed from A to C.

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

Well done.

For question two, Sofia and Jun are running a race.

The winner is the person who travels the furthest in 40 minutes.

Sofia runs at a speed of five kilometres per hour for 30 minutes, and then runs for 10 minutes at a speed of eight kilometres per hour.

Jun runs at a constant speed of six kilometres per hour for 40 minutes.

Who has won the race? And I want you to explain.

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

Well done, let's move on to question three.

Now question three says, the Oak teacher drove from Stoke to Bristol, she tabulated her journey but spilled some coffee over her table.

I want you to work out the missing information.

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

Well done.

Let's move on to these answers.

Well, for question one, you should have got these average speeds.

Massive well done if you've got this, press pause if you need more time to copy things down.

Really well done.

Let's have a look at question two.

Here's the working out, and hopefully you've realised that Jun won the race as he ran four kilometres in 40 minutes and Sofia ran 3.

8 kilometres, to one decimal place, in 40 minutes.

Well done if you got this.

For question three, once again, here's all the answers, massive well done if you got this one right.

Great work everybody.

So now it's time for the second part of our lesson, looking at solving problems involving speed.

Now speed is used every day in real life.

So when solving problems involving combining speed, we use our skills of combining speed with different representations, other topics of mathematics, and sometimes, really interesting real life facts.

For example, the speed of jets and fighter jets are measured using Machs.

And the Mach number describes the aircraft's speed compared with the speed of sound in air.

Now the Mach number is the ratio of flow velocity after a certain limit of the sound's speed.

And this is a basic table showing the range of speeds given the Mach number.

So a Mach number less than 0.

8 is less than 609 miles per hour, which is the same as being less than 980 kilometres per hour.

A Mach number between 0.

8 and 1.

3, including 0.

8, has this range of miles per hour and kilometres per hour.

A Mach number between 1.

3 and five, including 1.

3, has this range in miles per hour, which is exactly the same as this range in kilometres per hour.

And a Mach number between five and 10, including five, has the massive range in miles per hour and this in kilometres per hour.

You've probably heard of Mach numbers in some films whenever they're using jets or fighter jets, but now you can appreciate the range in speeds given the Mach number.

So if a jet started in London and was travelling at Mach 1.

3 for 10 minutes, and then Mach five for 10 minutes, how far would the jet have travelled? And we're asked to give our answer in kilometres.

From our table, you know Mach 1.

3 is 1,470 kilometres per hour.

Mach five is 6,126 kilometres per hour.

So let's see if we can work out the total distance travelled and that total time.

Well for Mach 1.

3, we know in 60 minutes it would've travelled a massive 1,470 kilometres, but how far would it have travelled in 10 minutes? Well, we divide by six, which tells me 245 kilometres were travelled in 10 minutes by our jet.

Now let's have a look at how far it travelled in Mach five.

Well Mach five, it travels 6,126 kilometres in 60 minutes.

So in 10 minutes, once again we divide by six, meaning, in 10 minutes the jet has travelled 1021 kilometres.

Therefore in 20 minutes, the jet has travelled 245, add our 1021, which gives us a huge 1,266 kilometres.

So that means as an example, in 20 minutes from London, the jet could be in Marseille.

In 20 minutes from London, the jet could be in Berlin.

In 20 minutes the jet could be in Spain, or as another example, in 20 minutes from London, the jet could be in Croatia.

That's an incredible speed.

So let's have a look at a quick check.

A standard aeroplane flies at an average Mach 0.

8, and it takes nine hours and 15 minutes to fly from London to Orlando.

Now if a jet plane left London and flew at Mach 1.

3 for 45 minutes and then Mach five, how long would the jet need to fly at Mach five for, to reach Orlando? I want you to give your answer in minutes to one decimal place.

Here you've got our table, see if you can give it a go.

Press pause, as you'll definitely need more time.

Well done, let's see how you got on.

Well, let's first of all work out the distance from London to Orlando.

Given the fact that we know nine hours and 15 minutes is equal to 555 minutes, and we know Mach 0.

8 is 980 kilometres per hour, 60 minutes.

We can work out the distance as the plane took 555 minutes to get from London to Orlando.

So converting our 60 minutes into 555 minutes means we multiply by our 9.

25, giving me 9,065 kilometres.

So the distance between London and Orlando is 9,065 kilometres.

From here, let's have a look at our jet plane.

We knew the jet plane travels at Mach 1.

3 for 45 minutes.

Mach 1.

3 has a speed of 1,470 kilometres per hour.

So to work out the distance in 45 minutes, I simply multiplied by 3/4, which means our jet plane travelled 1102.

5 kilometres in 45 minutes.

So now let's have a look at Mach five.

Well, we know at Mach five, it's a speed of 6,126 kilometres per hour.

Now we need to identify what distance does the jet plane need to fly in order to get to Orlando.

Well, it'd be our 9,065, the distance from London to Orlando, subtract our 1102.

5 kilometres, because that's the distance already travelled in the first 45 minutes, thus giving me a distance of 7962.

5 kilometres.

So I need to work out, how long does it take our jet plane at Mach five to travel 7962.

5 kilometres? Well to do it, I multiply by 1.

299 to three decimal places.

This gives me 78 minutes.

So that means the jet will need to fly for 78 minutes at Mach five after travelling 45 minutes at Mach 1.

3, incredible speeds again.

Fantastic work everybody.

So now it's time for your task.

Laura and Aisha are running the same race.

Aisha runs for 15 minutes at an average speed of four miles per hour.

She then runs for 20 minutes at an average speed of nine miles per hour.

If it takes Laura 40 minutes to run the same distance.

What is Laura's average speed in miles per hour.

See if you can give it a go.

Press pause for more time.

Well done.

Let's have a look at question two.

Saturn's moon, Titan, has been described as the most Earth-like celestial object in the solar system.

The spacecraft travels at a speed of 13 kilometres per second for five days, and then at a speed of 20 kilometres per second for the remainder of the journey.

Now of the distance from the Earth to Titan is 1.

2 times 10 of the power of nine kilometres.

How many days to the dearest day, will it take for the spacecraft to reach Titan from Earth? Great question, see if you can give it a go.

Press pause if you need more time.

Great work, let's move on to these answers.

Well, hopefully you realise, you need to work out the total distance of the race.

Given that Aisha runs for 15 minutes at an average speed of four miles per hour, this means it's a distance of one mile.

And then she runs 20 minutes at an average speed of nine miles per hour, which gives me a distance of three miles, so we now know the distance of the race is four miles.

So that means if Laura runs for 40 minutes at the same distance, Laura is running four miles in 40 minutes, and that's exactly the same as six miles per hour.

Really well done if you've got this one.

For question two, well first things first, let's work out how many seconds are in five days? Massive number, 432,000 seconds.

Now given that we're travelling at a speed of 13 kilometres per second, that means the spacecraft has travelled 5,616,000 kilometres in five days.

But we know the remaining distance has to be found.

So it's 1.

2 times 10 to the nine, subtract our 5,616,000, gives us a big number of 1,194,384,000 kilometres at a speed of 20 kilometres per second.

Therefore, we can work out the time.

So to travel that massive distance at a speed of 20 kilometres per second, means it'll take 59,719,200 seconds, which works out to be 691.

1944, et cetera, et cetera days.

Rounding up gives me 692 days.

Really well done if you got this.

Great work everybody.

So in summary, when asked to find the average speed, it's important to know the total distance travelled and the total time taken, as this will allow you to calculate the average speed.

And ratio tables can be helpful to structure the working out.

Speed is commonly used in real life, and when solving problems involving combining speed, we use our skills of combining speed with different representations, other topics of mathematics, and some really interesting real life facts.

I hope you've enjoyed the lesson today.

Well done, great work.