Lesson video

Hi, everyone, my name is Ms Ku, and today I'm really excited to be learning with you as we'll be looking at the wonderful unit ratio.

I hope you enjoy the lesson.

So let's make a start.

Hi, everyone, and welcome to this lesson on Checking and Securing Understanding of Simplifying and Unitising Ratios, and it's under the unit Ratio.

And by the end of the lesson, you'll be able to simplify a ratio, including expressing a ratio in unitary form.

So let's have a look at some key words.

A unitary ratio is a ratio where one part of the ratio is 1, and we'll be looking at that in our lesson today.

We'll also be looking at the words proportionality.

Remember, proportionality means when variables are in proportion if they have a constant multiplicative relationship.

Also remember a ratio shows the relative sizes of two or more values and allows you to compare a part with another part in a whole.

We'll also be looking at the highest common factor, and the highest common factor of two or more integers, which can be divided by all possible common factors.

HCF is an abbreviation for the highest common factor.

Today's lesson will be broken into two parts.

So let's make a start, simplifying ratios.

Now a ratio shows the relative sizes of two or more values and allows you to compare a part with another part in a whole.

For example, what is the same and what is different with the following ratios? See if you can give it some time.

Well, the first ratio shows T-shirts to shorts is 3:9.

The second ratio shows T-shirts to shorts is 2:6.

And the third ratio shows T-shirts to shorts is 1:3.

So the proportions are the same because for every one T-shirt, there are three shorts.

You can see it here and here, for one T-shirt, there are three shorts, for one T-shirt, there are three shorts.

You can also see it here, for one T-shirt, we have three shorts, for one T-shirt, we have three shorts, and for one T-shirt we have three shorts, but the total clothing is different.

So the proportions are the same but the numbers are different.

So all these ratios show the same proportion, so that means they're equivalent ratios.

However, there is only one ratio in its simplest form.

Do you think you can spot it? Well done.

Well, it's 1:3.

This ratio is in its simplest form, but how do we know? Well, the highest common factor of 1 and 3 is 1.

So when simplifying, the highest common factor of the integers must be equal to 1.

So that's how we can simplify a ratio.

So let's put them in a ratio table as this will allow us to see that multiplicative relationship easily and help us identify the simplest form.

I've popped my ratio in here, in here, and in here.

So we can spot the ratio 3:9.

Well, the highest common factor of 3 and 9 is 3.

So that means dividing both numbers by the highest common factor, which is 3, gives us the ratio in simplest form.

Let's have a look at the ratio 2:6.

Well, the highest common factor of 2 and 6 is 2.

So when we divide both of our numbers by 2, once again, it gives us the ratio in its simplest form, which is 1:3.

Therefore, we know the ratio of 1:3 is the ratio written in its simplest form as the highest common factor of the integers of 1 and 3 is 1.

Now it's time for your check.

Here are four bar models showing bananas and apples.

I want you to write each as a ratio and then identify the ratio in its simplest form for them all.

See if you can give it a go and press Pause if you need more time.

Well done.

So let's see how you got on.

Well, the first bar model shows the ratio of apples to bananas to be 6:9.

The second bar model shows the ratio 4:6.

The third bar model shows the ratio of 2:3.

And the last bar model shows the ratio of 1:1.

5.

So now we have all of our ratios, let's see if we can identify the simplest form for all of them.

Well, here you can see our our four equivalent bar models shown in a ratio table, and putting 'em in a ratio table really does help us identify that ratio in its simplest form.

Let's start looking at the ratio 6:9.

Well, the highest common factor is 6 and 9 is 3.

So dividing the 6 and the 9 by the highest common factor gives us the ratio of 2:3.

Now let's have a look at the ratio 4:6.

Well, the highest common factor of 4 and 6 is 2.

So dividing 4 and 6 by 2 gives us the ratio of 2:3.

So that means the ratio of 2:3 is the ratio written in its simplest form as the highest common factor of the integers is 1.

Laura says, "Why isn't the answer 1:1.

5? As this ratio is equivalent and it's got the smallest numbers?" Do you think you can explain to Laura why the simplified ratio cannot be 1:1.

5? Well done.

Well, it's simply because the ratio in its simplest form must be made of integers and have a highest common factor of 1.

So that means we can simply multiply each part of the ratio of 1:1.

5 by 2, giving us the simplified ratio of 2:3.

Well done if you've got this.

So you can show your working to simplify a ratio with or without a ratio table.

For example, asked to write 40:25 in its simplest form.

Now, we can put this in a ratio table.

So let's call 40 A and 25 B, or we can simply keep it as a ratio 40:25.

Looking at the highest common factor of 40 and 25, we know it's 5.

So with or without our ratio table, we simply dividing by 5.

Therefore, we know 40:25 written in its simplest form is 8:5.

Well done if you spotted this.

Now it's time for another check.

Sofia and Sam are given the ratio 40:20:60 to write in its simplest form.

Can you explain the differences in their methods? 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, Sam has used a ratio table.

And divided by a factor of 10 first, then divided by a factor of 2.

Sofia hasn't used a ratio table but has divided each number by the highest common factor, which is 20.

Both methods are correct, but Sofia's method using the highest common factor is more efficient.

So just remember, you can simplify a ratio in steps, or identifying the highest common factor allows you to identify the simplest ratio in one simple step.

Well done if you've got this.

Now it's time for another check.

What I want you to do is write the following ratios in their simplest form.

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, for A, writing the ratio in a ratio table, you could have spotted that the highest common factor is 10.

So dividing the 40 and the 50 both by 10 gives us the ratio 4:5.

18:24, gonna pop it in my ratio table and identify the highest common factor is 6.

So dividing both numbers by 6 gives me the ratio of 3:4.

2:2.

5, while you might spot, we do not have an integer here.

So simply by multiplying by 2, I have two integers which are 4:5.

I've also identified that the highest common factor of 4 and 5 is 1.

So that means my ratio in its simplest form is 4:5.

Next, we have 10:30:15.

Exactly the same, I'm gonna pop it in a ratio table, identify the highest common factor, which is 5.

And then from here, and identify the ratio to be 2:4:3.

Really well done if you got this.

Now let's have a look at another check.

Jacob simplified the ratio 10:20:15 and got the ratio of 1:2:3.

Jacob said, "2 and 4 have a common factor of 2, so I divided both of those by 2." Do you agree with Jacob? 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, unfortunately, Jacob is incorrect.

To simplify a ratio further, all of the parts must share a common factor, and every part must be divided by that factor.

So that means 2:4:3 is the simplest form of the ratio 10:20:15.

Now furthering this a little bit more, Lucas and Izzy are given these ratios and asked to find the odd one out.

Lucas says, "The odd one out will be the one where the multiply between the parts will be different." And Izzy says, "When writing them in their simplest form, we can spot the odd one out." Who do you think is correct? Well, both of them are correct.

Either method can be used.

So let's have a look.

Well, we're gonna look at Lucas's method first.

Putting them in a ratio table can help us spot that multiplicative relationship a little bit easier.

So popping them into a ratio table, I have this.

Now let's identify that multiplier.

Well, if we multiply 8 by 2.

5, it gives us 20.

If we multiply 20 by 2.

5, it gives us 50.

If we multiply 12 by 2.

5 it gives us 30.

But if we multiply 18 by 3, that's how we get our 54.

So that means we know 18:54 is the odd one out.

Now let's have a look at Izzy's method.

Izzy wants us to put them in the simplest form and then we can spot the odd one out.

So let's have a look at the ratio 8:20.

Well, the highest common factor of 8 and 20 is 4.

So dividing the 8 and 20 by 4 gives us the simplified ratio 2:5.

20 to 50.

The highest common factor is 10.

So dividing both numbers by 10 gives us the ratio of 2:5.

12:30.

The highest common factor is 6.

So dividing both numbers by 6 gives me 2:5.

Lastly, 18:54.

The highest common factor is 18.

So dividing both numbers by 18 gives us 1:3.

So that means once again we know 18:54 is the odd one.

Out of curiosity, which one do you prefer? Have a little think.

Well, Lucas's method is good, but sometimes the multiply between parts isn't always a simple number.

The good thing about Izzy's method is that any ratio can be written in its simplest form.

What I want you to do now is another check.

Which of the following is the odd one out? And I want you to explain why.

See if you can give it a go.

Press Pause if you need more time.

Well, 24:42 is the odd one out.

I'm going to choose simplifying each ratio.

Simplifying each ratio, you might spot they all simplify to 3:5, except for 24:42, which simplifies to 4:7.

Really well done if you've got this.

So when writing a ratio in its simplest form, we recognise the ratio to parts must be integers and have a highest common factor of 1.

However, there is also something else to consider, which is very important.

For example, 4 centimeters:5 metres is not in its simplest form.

Can you explain why? Well, we must also consider the units.

The units must be the same in order to simplify.

So let's have a look at 4 centimeters:5 metres and convert them both to centimetres or both metres.

In order to avoid decimals, I'm going to find it easier to convert both of these units into centimetres.

So that means I have 4 centimeters:500 centimetres.

Now given the fact that the units are the same, we can simply remove the units and put them in a ratio table.

So we have 4:500.

Then simplify using the highest common factor.

Well, the highest common factor of 4 and 500 is 4.

So this means 4 centimeters:5 metres in its simplest form is 1:125.

Well done.

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

I want you to match each of the following to the correct scale.

I've given you a little hint and given you some unit conversions there.

See if you can give it a go.

Press Pause if you need more time.

Well, to avoid decimals, I'm just going to convert to centimetres.

So it's 3:1,500, which simplifies to 1:500, which is matched here.

Next, 6 centimeters:3 kilometres.

Then again, I'm gonna convert to centimetres to avoid decimals.

This gives me 6:300,000, which simplifies to 1:50,000, which gives me my answer here.

Next, 5 centimeters:250 metres.

Converting them both to centimetres, I can match it up to the ratio of 1:5,000.

And 4 centimeters:20 kilometres, big number in our ratio there is 1:500,000.

Really well done if you got this.

Great work, everybody.

So now it's time for your task.

What I want you to do is write the following ratios in their simplest form.

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 the shaded to non-shaded in their simplest form.

See if you can give it a go.

Press Pause for more time.

Well done.

Let's move on to question three.

Which of the following is the odd one out? See if you can give it a go and press Pause if you need more time.

Well done.

Question four wants us to write the ratios in their simplest form.

Notice how I've given you a bit of a hint with those unit conversions.

Press Pause for more time.

Great work, everybody.

This last question is really tough.

I'll be really impressed if you get this.

Aisha is asked to simplify the ratio 360:84:9,900.

She does this correct working out.

Can you help Aisha write the simplified ratio? See if you can give it a go.

Press Pause for more time.

Well done.

Let's go through these answers.

Press Pause if you need more time to mark it.

Massive well done if you've got these.

For question two, you should have the shaded to unshaded in its simplest form is 5:4.

And here, the shaded to unshaded is 11:7 in its simplest form.

Well done.

For question three, here are odd ones out.

Very well done if you got this.

For question four, using those unit conversions, we should have these ratios in their simplest form.

Well done.

Lastly, for question five, hopefully you spotted in the centre of our Venn diagram is the highest common factor.

So we simply divide each of our numbers by that highest common factor to give us the simplified ratio to be 30:7:825.

Really well done if you've got this.

Great work, everybody.

So now let's move on to writing a ratio in unitary form.

Now sometimes writing a ratio in its simplest form is not the most useful ratio to use.

For example, here are some equivalent ratios of a floor plan where the ratio shows the plan to real life.

We have 5:82, which is exactly the same as 10:164, which is exactly the same as 1:16.

4.

Explain why these ratios are all equivalent and which one is in its simplest form.

See if you can give it a go.

Well done.

Well, hopefully you've spotted that using a ratio table, you can see that multiplicative relationship and identify that they are all equivalent.

Now, we can also identify the fact that 5:82 is the ratio in its simplest form, but let's say you wanted to find out what a two centimetre plan length would be in real life.

Which ratio do you think would be more efficient to use? Well, it'll be this one because we simply need to multiply the plan length by 2 and 16.

4 by 2.

This means that the plan length in two centimetres is 32.

8 centimetres in real life.

So notice how we did not use the ratio in its simplest form.

We use the ratio where one part is given, and this is called a unitary ratio, where we write a ratio in the form of 1:n or n:1.

And this is because it allows us to calculate answers to calculations more efficiently.

For example, let's have a look at a fruit bar, and a fruit bar is made using the following ratio, dried fruit to syrup is in the ratio of 8:5.

The question wants us to write the ratio in the form of 1:n.

And then it's asks, if there were 36 grammes of dried fruit, how many grammes of syrup is needed? So to do this, a ratio table will help.

I'm gonna pop my dried fruit to syrup in a ratio table.

Now because part A wants us to write the ratio in the form of 1:n, I'm going to divide the 8 by 8 to give me 1, and that means I have to divide the 5 by 8 to give me 0.

625.

So that means my ratio is 1:0.

625.

It's not in its simplest form but it's in unitary form.

So if we were asked to work out 36 grammes of dried fruit, how much syrup is needed? To do this, we simply multiply the one part of dried fruit by 36.

This will give us 36 grammes.

That means we need to multiply the parts of syrup by 36 to give us 22.

5 grammes of syrup.

So notice how that unitary form really did help us work out that calculation quite quickly.

But now the question wants us to write the ratio in the form of n:1.

So the syrup must be one part.

Well, to do this, we're going to divide both sides by 5 because we need the syrup part to be one.

So dividing both side by 5 gives us 1.

6:1.

Notice again, this is in unitary form.

So now the question wants us to look at 41 grammes of syrup, how much dried fruit is needed? So all we need to do is multiply that one of our syrup by 41 and then multiply the 1.

6 by 41 gives us 65.

6 grammes of dried fruit as needed.

Once again, notice how writing in a unitary form allowed us to quickly calculate this answer.

So what I'd like you to do as a quick check is to write the following ratios in the form of 1:n, where n is a decimal.

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, for A, you should have got 1:4.

5.

This is because you should have divided both parts by 2.

For B, you should have got 1:1.

2.

This is because you should have divided both parts by 5.

For C, you should have got 1:1.

1.

This is because you should have divided both parts by 10.

Well done if you got this.

The second part, we need it in the form of n:1.

This means it should be 0.

75:1.

You should have divided both sides by 4.

The next ratio should have been 1.

4:1, dividing both sides by 5.

And the next ratio is 0.

05:1.

What you should have done is divide both parts by 20.

Really well done if you've got this.

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

Laura and Alex both write the ratio 0.

3:2 in the form of 1:n.

You can see the working out from Laura and you can see the working out from Alex.

Both methods are correct, but what I want you to do is explain the different methods each pupil has chosen.

See if you can give it a go.

Press Pause if you need more time.

Well, let's have a look at Laura's first.

Well, Laura's dividing each part by 0.

3.

In doing so, it's given the ratio 1:n.

She's then written note 0.

3 as a fraction.

And then she uses the fact that dividing a number is the same as multiplying by the reciprocal, thus giving the ratio of 1:20/3.

Now let's have a look at Alex.

Alex decided to write an equivalent ratio with integers only.

So as multiplied 0.

3 and 2 by 10.

And then from here, has divided each part of the ratio of our 3:20 by 3.

So it's in the form of 1:n.

Then Amex still gets exactly the same ratio, 1:20/3.

Really well done if you spotted the differences in their methods.

Out of curiosity, which one do you prefer? Great work, everybody.

So now it's time for your task.

What I want you to do is write the following ratios in the form of 1:n.

The second part, we need it in the form of n:1, where n is a decimal.

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 the area for shaded to unshaded in the form of 1:n.

See if you can give it a go.

Press Pause for more time.

Great work.

Let's have a look at question four.

Question four wants you to write the following in the ratio of 1:n.

The second part, we need it in the form of n:1, but n is a fraction.

See if you can give it a go.

Press Pause as you'll need more time.

Great work.

Let's move on to our last questions.

Jun answered 80% of the questions in his test correctly.

Write the ratio of correct incorrect in the form of n:1.

And lastly, a travel company offered that for every four pounds, you get $10.

Write this in the ratio of pounds:$1 See if you can give it a go.

Press Pause for more time.

Great work.

Let's move on to these answers.

Press pause if you need more time to mark.

Question three, you should have the ratio is 20:44, but in the form of 1:n, it should be 1:2.

2.

Next, you should have 40:24.

But in the ratio of 1:n, you should have 1:0.

6.

For question four, we should have all these wonderful ratios in the form of 1:n or n:1.

Moving on to question six, here, we should have the following answers.

Well done.

Great work, everybody.

So a ratio is in its simplest form when the integer parts of the ratio have a highest common factor of 1.

And writing a ratio in its simplest form allows you to see the proportions more easily.

For example, the equivalent ratios of 3:9 is 6:18 and 15:45, as they all simplify to 1:3.

Unitary form or writing a unitary ratio is when we write a ratio in the form of 1:n or n:1, as it allows some calculations to be completed more easily.

And given there are an infinite number of equivalent ratios, there are many different ways to use the multiplicative relationship to write a ratio in the form of 1:n or n:1.

Great work, everybody.

It was wonderful learning with you.