Lesson video

Hi everyone.

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

Today, we're going to be looking at bounds, upper bounds, and lower bounds and lots of key vocabulary in between.

You may know some of these keywords, you may not, but I'll make them clear during the lesson.

Great to be learning with you, let's make a start.

Hi everyone and welcome to this lesson on using upper and lower bounds, practically under the unit rounding, estimation and bounds.

By the end of the lesson, you'll be able to consider a context and decide whether it's suitable to use the upper or lower bounds for each value in a calculation.

Our keywords are upper bound and lower bound.

Remember, the upper bound of a rounded number is the smallest value that would round up to the next rounded value.

For example, six has been rounded to the nearest integer, means 6.

5 is the upper bound.

Another example would be 6.

3 has been rounded to one decimal place.

This means 6.

35 is the upper bound.

The lower bound for a rounded number is the smallest value that the number could have taken prior to being rounded.

We'll also be looking at the word error interval, and the error interval for a number X shows the range of possible values of X and it's written as an inequality where A is less than equal to x, which is less than B.

Today's lesson will be broken into three parts.

We'll be looking at using bounds in safety calculations first.

Then moving on to using bounds in budgeting and finance, and then using bounds with compound measures.

So let's make a start using bounds in safety calculations.

Now the degree of accuracy determines the upper and lower bounds of a number or measurement.

And when considering safety, it's sometimes necessary to refer to the upper bound or the lower bound, depending on the context of the question.

For example, when looking at the maximum capacity of a lift or a bridge or a cardboard box, would you consider the upper or lower bound of the maximum capacity? Have a little think.

Well, for safety reasons, considering the lower bound of the capacity is the safest.

Now what we're going to do is have a look at the masses or items. Well, if we're looking at the maximum capacity of a lift, bridge or cardboard box, would you consider the upper or lower bounds of the masses or items which go on or inside? Have another think.

Well, for safety reasons, considering the upper bound of these masses is the safest.

For example, the upper bound of the cars or people, or items which go in the box.

So let's have a look at an example.

A box can hold a maximum mass of 20 kilogrammes to the nearest kilogramme, and Aisha has the following items. She has a mid-tower gaming desktop, which is 10 kilogrammes measured to two significant figures.

A PC monitor, which is three to the nearest integer, a remote control robot, which is four kilogrammes to one significant figure and a sound system, which is six kilogrammes to the nearest kilogramme.

What combination of items can she not confidently and safely put in the box, so not to exceed the maximum mass? Have a little think.

Well, first of all, let's identify the error interval of what we have.

The error interval of the capacity of the box is given here, the error interval of our tower, the error interval of our monitor, the error interval of our robot and the error interval of our sound system.

Now for confidence and assurance that the box will not exceed maximum mass, it's best to use the lower bound of the capacity of the box and the upper bound of each item.

So considering these, she must not use the following combinations, the tower, the monitor, and the speakers, because when you sum those upper bounds, they give 20.

5 kilogrammes which exceeds the lower bound of the capacity of the box.

Another combination which you shouldn't use is the tower, the robot and the speakers because summing those upper bounds gives 21.

5 kilogrammes, which once again exceeds the lower bound of the capacity of the box.

So these are the combinations that she must not put into the box as the sum of these upper bound masses exceeds the lower bound of the capacity of the box.

Now let's have a look at a check question.

A lorry needs to pass over a weak bridge to transport some cement, and the sign states that the maximum gross weight of 7.

5 tonnes is allowed on the bridge.

The lorry weighs six tonnes to the nearest tonne and the lorry needs to transport 30 50 kilogramme bags of cement each measured to the nearest kilogramme.

Show that the lorry cannot safely and confidently transport the cement in one trip.

See if you can give it a go.

Press pause if you need more time.

Well done.

So firstly, let's identify our error intervals and ensure that all the units are the same.

So let's look at our lorry first.

Here is the error interval for the mass of the lorry given in kilogrammes.

Now that's identified the error interval for our 30 50 kilogramme bags of cement.

Well, to do this, we multiply 30 by the lower bound of the mass of each bag, which is 49.

5 kilogrammes, and to work out the upper bound of the cement, it would be 30 multiplied by the upper bound of the mass of each bag of cement, which is 50.

5.

Working this out means we have the error intervals for our lorry, and the error intervals for our 30 bags of cement.

So solving the upper bounds of our lorry and the upper bounds of our cement, it does exceed 7,500 kilogrammes.

In other words, it exceeds 7.

5 tonnes.

So that's why, it's not safe for the lorry to transport all 30 50 kilogramme bags of cement in one trip.

Well done if you got this.

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

Read the question carefully.

Ensuring when to use the upper bound or the lower bound to identify the maximum number of boxes you can put in a lift and to identify why would this maximum be dangerous.

See if you can give it a go.

Press pause if you need more time.

Great work, let's move on to question two.

You need to explain the relationship between precision, degrees of accuracy and error bounds.

Take your time.

This is a nice question where involves you to write a few sentences using these keywords so you can give it a go.

Press pause for more time.

Well done, let's look at these answers.

Well, for question one, the lift has a maximum capacity of 800 kilogrammes measured to the nearest 10 kilogramme.

So identifying the error interval for the lift is given here.

We have a box where each box is eight kilogrammes measured to the nearest kilogramme, so therefore the error interval for the box is given here.

So that means to work out the maximum number of boxes, it would be, the upper bound of the lift divided by the lower bound of the boxes giving us 107 boxes.

So the maximum number of boxes that he could put in the lift is 107 boxes.

But why would this be dangerous? Well, it'd be dangerous because if Andeep puts 107 boxes into the lift, and each box has this error interval, the potential mass of all the boxes could be given as 802.

5 kilogrammes less than equal to the mass, less than 909.

5 kilogrammes.

In other words, you can see that the lower bound of the 107 boxes, exceeds the maximum capacity of the lift.

So it wouldn't be a good idea to calculate the maximum number of boxes that could fit into the lift.

For question two, explain the relationship between precision, degrees of accuracy and error bounds.

Well, if a measurement was precise, then there would be no degrees of accuracy and there'd be no error bounds.

But it is impossible to have a precise measurement as measurements are continuous.

Remember, every measurement will have some sort of degree of accuracy.

The more accurate, the less difference between the upper and lower bounds, and the less difference, the more accurate the measurement is.

Well done if you've got this.

Great work, everybody.

So now let's move on to the second part of our lesson using bounds in budgeting and finance.

In a 2023 report, a trade body British retail consortium estimated that £953 million had been lost to retail theft.

However, top retailers in 2023 in the UK still recorded record profits.

So how can they make huge losses but also make such good profits? Well, it's because strong businesses take into account areas where losses occur and counterbalance this with effective budgeting and financial planning.

Sometimes it's best to focus on the lowest possible profit as a minimum basis as this allows a contingency for other the unexpected factors.

For example, looking at retail and negative contributing factors such as theft, accidental damage, et cetera.

Other times it's best to consider the highest cost when budgeting, for example, renovating a house, the range of expected costs for each type of renovation.

Now let's have a look at an example.

In a factory, a machine fills one kilogramme bags of flour correct to the nearest 10 grammes and it costs £1.

05 to produce and the factory sells each bag for £2.

54.

If a machine can fill 30 bags per minute, what is the minimum amount of flour that might be weighed out in an hour? Well, first of all, we have different units, so let's convert them all to the same units.

We know that the bag of flour is one kilogramme correct to the nearest 10 grammes.

So converting this all into grammes means we have 1000 grammes plus or minus five grammes, which gives us the error interval of each bag of flour to be this.

Now from here, if we know 30 bags of flour are produced every minute, we simply multiply this by 30, giving this error interval for how many bags are made per minute.

Now you can see the minimum is seen as the lower bound of the amount.

So that means we know the lowest amount of flour that will be weighed out per minute is 29.

850 kilogrammes of flour per minute.

But, given the question once it has per hour, we simply multiply this by 60, meaning it's 1791 kilogrammes of flour per hour.

This would be the minimum amount of flour needed per hour.

Now, if the factory has 600 kilogrammes of flour correct to the nearest five kilogrammes, what's the minimum profit that could be made from the total sales? Well, same again, let's identify that error interval of the flour.

We have kilogrammes and grammes converting it all to the same unit.

Now we have the error interval for our 600 kilogrammes of flour.

I've labelled this as a capital F.

Now we know the error interval for each bag of flour from before, I've labelled this as lowercase F.

Now converting to the same units means we're able to do our calculation more effectively.

So how would we calculate the minimum number of bags that could be made using this 600 kilogrammes of flour? Well, to do this, it would be the lower bound of the 600 kilogrammes of flour, divided by the upper bound of what's required to make a bag of flour.

Working this out, gives me 594.

52736, which means the minimum number of bags of flour to be produced is 594.

So to work up the profit, it's the difference between the selling price and the cost to produce, multiply by our 594.

This gives us a minimum expected profit of £885.

06.

Now, let's have a look at a check question.

A carpenter buys a six metre correct to the nearest 10 centimetres, length of wood for £10 He cuts it into 40 centimetre blocks where each 40 centimetres is correct to the nearest 10 centimetres, and he sells each block for £1, which calculation shows the minimum profit? Have a little think, write some things down if you need and press pause if you want more time.

To work out the minimum profit, the lower bound of the six metre length need to be divided by the upper bound of the size of each block, making sure you convert to the same units.

You then multiply this number by £1 and then obviously subtract the cost of wood, this will give you the profit.

Well done if you got this.

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

Read the question carefully.

Make sure you convert to the same units to make things easier, and then work out how much should she budget, so to tile her wall.

so you can give it a go.

Press pause if you need more time.

Well done.

Let's have a look at question two.

Question two, same again, make sure you look at those units carefully and work out if there are 20 kilogrammes of coffee beans to the nearest kilogramme, what is the minimum profit expected? See if you can give it a go.

Press pause if you need more time.

Great work.

Let's go through these answers.

Well first of all, let's work out the error intervals of the wall and tile and then let's work out to get the maximum cost of our tiles.

It's £732.

55 If you want to press pause and have a look at any of this working out, please do.

Now we can work out the total cost of the wall by simply doing the maximum cost of the tiles, and the maximum cost of the adhesive.

Working this out, we can work out the total maximum cost to tile the wall to be £907.

86 Great work, if you've got this one.

For question two.

Same again, working out those error intervals really does help us out.

Then we can identify the minimum profit to be £205.

20, press pause if you need more time to have a look at this working out.

Well done.

Great work, everybody.

So now let's have a look at using bounds with compound measures.

The application to bounds in real life is so important for safety reasons, accuracy in engineering and medicine, sporting competitions, and so much more.

Therefore, applying the correct bounds into a formula, so to get the maximum or minimum value is equally important.

For example, if we were asked to calculate speed, this is a common formula.

Speed is equal to distance over time.

What's really important to do, is to pay attention to our units whenever we're using formula.

In this example, Aisha jogs 40 metres to the nearest metre and she jogs it in 20 seconds to the nearest second and she says, "My speed is exactly two metres per second." Is she correct? Have a little think.

Well given the distance and time have been rounded, this means it will not give her the exact speed.

But what we can do, is find the error interval for our speed.

So let's work out the error interval for our speed.

Firstly, we need to work out the error interval for distance, we know that the distance was 40 metres measured to the nearest metre, so this is the error interval for our distance.

Then let's work out the error interval for our time.

We know it was 20 seconds to the nearest second.

So here's our error interval for our time.

Then to work out the lower bound of our speed, it would be the lower bound of the distance divided by the upper bound of our time.

So dividing these gives us the lower bound to be 1.

93 seconds, to three significant figures.

To work out the upper bound of the speed, it would be the upper bound of the distance divided by the lower bound of the time.

This is 40.

5 divided by 19.

5, giving us 2.

08 seconds to three significant figures.

So now, we have the error interval for our speed.

1.

93 metres per second is less than S, which is less than 2.

08 metres per second.

So although Aisha didn't calculate her exact speed, because she rounded, we have an error interval for her speed.

Now let's have a look at a check.

Izzy is competing, and she knows the best runner can run 10 kilometres to the nearest kilometre in 50 minutes to the nearest minute.

Now Izzy reckons she can run at a constant speed of 12 kilometres per hour.

Does Izzy have a chance to beat the best runner? See what you can work out.

Take your time and press pause.

Great work.

Let's see how you got on.

Well, first of all, let's convert to the correct units for the best runner.

So given the fact that Izzy's speed is kilometres per hour, we have to convert it into hours.

So I'm going to simply divide her time by 60, giving us the time in hours.

Now from here, the best runner's speed has this following error interval, substituting in what we know, the speed of the best runner to be in between 11.

3 kilometres per hour and 12.

7 kilometres per hour.

So given that Izzy reckon she can run at a constant speed of 12 kilometres per hour, there is a chance that she could win, as 12 kilometres per hour is greater than the lower bound of the best runner.

Well done if you got this.

Great work everybody.

Now let's have a look at your task, read the questions carefully and pay attention to those units.

Press pause if you need more time.

Well done.

So let's have a look at these answers, looking at a, first.

Given the fact that Laura runs five kilometres to the nearest kilometre in 30 minutes, correct to the nearest five minutes, and Sophia can run 11 kilometres per hour, correct to the nearest kilometre per hour.

If both pupils were to race each other, is there a chance that Laura could win? Let's look at those units.

Well, first of all, identifying the error interval for the speed of Laura means we have to look at converting those minutes into hours.

So you can see the working out here.

From here, we have our error interval of the speed of Laura to three significant figures.

We know it's in between 8.

85 kilometres per hour and 11.

19 kilometres per hour.

Let's have a look at the error interval for Sophia, or the error interval for Sophia goes from 10.

5 kilometres per hour to 11.

5 kilometres per hour.

So there is a chance that Laura could win as the upper bound of Laura's speed, is greater than the lower bound of Sophia's speed.

But what assumptions have been made in these calculations? We are assuming that both girls run at a constant speed.

Now Andeep says if Laura runs 44 kilometres per hour, she would certainly win.

Explain if Andeep is correct.

Well, yes, Andeep is correct.

If Laura were to run 44 kilometres per hour, she would most certainly win the race.

But given the fastest person to date is Usain Bolt in 2009, ran 9.

58 metres per second, this is approximately the same as 34.

5 kilometres per hour.

So therefore, it's incredibly unlikely that she'll run a 44 kilometres per hour.

Great work everybody.

So in summary, the degree of accuracy determines the upper and lower bounds of a number or measurement.

And when considering safety, it's sometimes necessary to only refer to upper bounds or lower bounds, depending upon the context of the question.

The application to bounds in real life is so important for safety reasons, accuracy in engineering and medicine, sporting and competitions, and so much more.

Therefore, applying the correct bounds into the formula, so to get the maximum or minimum value is equally important.

Great work everybody, it was wonderful learning with you.