Lesson video

Hi everyone.

Welcome to our lesson.

Today, we'll be looking at multiplicative relationships.

It's one of my favourites, because it appears so much in everyday life.

Really excited that you're learning with me today, so let's make a start.

Welcome to this lesson on expressing one number as a percentage of another, and it's under the unit understanding multiplicative relationships, percentages and proportionality.

By the end of the lesson, you'll be able to describe one number as a percentage of another.

We'll also be recapping on the word equivalent.

Now two fractions are equivalent if they have the same value.

For example, 1/2 is exactly the same as 2/4.

4/5 is exactly the same as 40/50.

A non example would be 1/5 is not the same as 3/7.

Today's lesson will be broken into two parts.

First, will be writing any fraction as a percentage, and then we'll be writing any number as a percentage of another.

So let's make a start writing any fraction as a percentage.

Now the word percent comes from a Latin phrase per centum, meaning by the hundred and a hundred square is a helpful way to show a percentage.

As the total represents 100%.

Given decimals, percentages and fractions, all show proportion, we're able to interchange between them.

So what I want you to do is think, what do you think 1% looks like on this hundred grid? Well, hopefully you spotted when it's only one square filled in, this represents 1%.

And it could be any one square filled in as it still represents 1%.

Now, what do you think 1% is as a fraction? Well, 1% simply means 1 per 100 or 1/100.

So we're able to recognise the relationship between percentages and fractions.

Now what I want you to do is try this check question, what percentage and fraction is shaded here on our hundred grid? Take your time and press pause if you need more time.

Great work.

Let's see how you got on.

Well, hopefully you've counted there are 27 squares, which make that word hi.

So it's 27%.

Now given that each square represents 1% and we have 27 squares shaded, then we can clearly see it's 27%, but as a fraction it's 27/100, because it's 27 shaded out of 100.

Well done if you got this one.

But what happens when we don't have a hundred grid? What percentage do you think is shaded here? Well, first of all, let's write it as a fraction.

Well, we know it's 17 shaded out of 20.

So what I'm gonna do is think about it as bars.

For every 20 bars there are 17 shaded and we need to keep the proportion the same but make a hundred grid.

So 17 shaded out of 20 bars.

That means there would be 34 shaded out of 40 bars.

We still have the same proportion, 17 is shaded out of 20.

We have equivalent fractions.

Now it's also the same as 51 shaded out of 60.

We still have 17 shaded out of 20 bars.

These are equivalent.

So that means it's equivalent to 68/80.

The number of shaded to non-shaded is still in proportion, and that means we now have 85 shaded out of 100.

So we've converted our 17 shaded per 20 bars into an equivalent fraction of 85 per 100.

So that means my percentage is 85%.

Knowledge of equivalent fractions allows us to make the denominator of the fraction 100 to work out the percentage.

So making the denominator of the fraction 100 allows us to change to a fraction and to a percentage easily without the need of hundred grids.

For example, let's change 16/25 as a percentage.

We need to make that denominator 100.

So what we're going to do is we're going to multiply by the fraction 4/4.

Multiplying 16/25 by 4/4 is the same as multiplying 16/25 by 1.

Now using our knowledge on multiplication of fractions, this means we have 16 multiplied 4 over our 25 multiplied by 4, thus giving us a denominator of 100, which is what we wanted, and a numerator of 64.

So that means my percentage is 64%.

And what we've successfully done here is we've converted our fraction into a percentage without using a hundred grid.

Now what I'm going to do is I'm going to do a question and then I'd like you to try another question on your own.

We're going to convert 13/20 as a percentage.

Well 13/20, we want that denominator to be 100.

So I'm going to multiply by 5/5, thus using our knowledge or multiplication of fractions, 13 multiplied by 5 over 20 multiplied by 5, thus giving me 65/100.

And remember we wanted that denominator of 100, so we can convert it into a percentage giving me 65%.

Now what I'd like you to do is write 31/50 as a percentage.

Ensure to show all you're working out.

Great work.

So let's see how you got on.

Well, 31/50, if we multiply by 2/2, this will give me 31 multiplied by 2 over 50 multiplied by 2.

I now have a denominator of 100, which is what I wanted, thus giving me a percentage of 62%.

Really well done if you got this one right.

So what I want you to do now is using those skills, I want you to convert the following fractions into percentages and I'd like you to show your working out.

So 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, 7/10, we have to multiply by 10/10, thus giving us 70/100, which we know is 70%.

1/5 multiplied by 20/20, giving us 20/100, which is 20%.

Now 17/200, we still want to make that denominator 100, so we're going to divide by 2/2, thus giving us 8.

5/100, which is 8.

5%.

12.

3/300, we still want that denominator to be 100, so we're going to divide by 3/3, thus giving us 4.

1/100, which is 4.

1%.

Those last two were really hard, so very good if you got that one.

Now Alex says, "But what about fractions where the denominator can't be easily changed into 100? For example, 1/12." And Sophia says, "Good point, what do we do?" Well, let's look at a hundred square grid first and see if we can find another way.

Looking at this hundred square grid, we know as a percentage this represents 1%, as 1 is shaded out of our 100.

But now I'm going to do is change this hundred grid and make the whole hundred grid represent one whole.

So this whole grid represents one whole, what would this little square be as a decimal if we know the whole thing represents one? See if you can give it a bit thought.

It has to be 0.

01, and this is because if you have 100 0.

01's, they would work out to be 1.

So this means 0.

01 is equivalent to 1%.

Using this fact, we can convert any fraction into a decimal and then a decimal into a percentage.

So knowing that 0.

01 is equivalent to 1%, we can convert any fraction to a decimal which can then be converted into a percentage.

So let's have a look at that question that Alex mentioned.

We need to change 1/12 into a percentage.

Well, to do it, I'm going to use short division as we can work out the decimal of 1/12.

1/12 simply means 1 divide by 12.

Using short division, we put the divisor of 12 outside and the 1 inside, how many twelves go into 1? Well, it's none.

Don't forget your decimal point and there's trailing zeros.

Remember, we have a remainder of 1, and then we ask ourselves, how many twelves go into 10? Well, it's none.

Then we have another trailing zero with a remainder of 10.

So how many twelves go into 100? Well, it's 8 remaining 4.

With another trailing zero, how many twelves go into 40? Well, it's 3 with a remainder 4, then with another training zero how many twelves go into 40? Well, it's 3, so on and so forth.

So what have we just found out? We've just found out that 1/12 is exactly the same as 0.

83 recurring.

But how do we change this into a percentage? Well, change it into a percentage, remember, we know 0.

01 represents 1%.

So knowing that that means 0.

083 recurring represents 8.

3% rounded to one decimal place.

Short division is a really nice way to identify the equivalent decimal of our fraction.

Well done, so let's have a look at a quick check.

What I want you to do is identify what is 1/3 as a percentage to one decimal place.

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, 1/3 basically means 1 divide by 3.

So using our short division, 3 is our divisor, so it's outside and 1 is on the inside.

How many threes go into 1? Zero, don't forget your decimal point, and those trailing zeros.

We have a remainder of 1, how many threes go into 10? 3.

What's remaining? 1, and our trailing zero.

How many threes go into 10? 3 remaining 1, so on and so forth.

So you can see we have a recurring decimal.

So now we know 1/3 is equivalent to 0.

3 recurring.

What is this as a percentage? Well, 1/3 as a percentage is 33.

3%.

Remember, 0.

01 represents 1%.

So that means we're able to convert 0.

3 recurring into 33.

3%.

Massive well done if you got this one right.

Now let's have a look at another check.

Three students convert 2/5 into a percentage, who is correct? Have a good look and see if you can work out who's correct.

Press pause for more time.

Great work, let's see how you got on.

Well, hopefully you spotted both Alex and Andeep are correct.

Sam just simply needed to multiply 0.

4 by 100 in order to convert to a percentage.

Multiplying the decimal equivalent by 100 is a very efficient way to convert a decimal into a percentage.

Very good.

Now let's move on to your task.

I only want you to shade in the equivalent percentages of the following fractions.

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

Great work, let's move on to question 2.

Question 2 wants you to convert the following fractions into percentages and give your answers correct to one decimal place if needed.

See if you can give it a go and press pause for more time.

Really well done, let's move on to these answers.

Well, hopefully you spotted these are our equivalent percentages.

So shading these in, what have we revealed? Well, we've revealed a lovely little percentage sign.

Great work if you've got this one right.

For question 2, let's look at a and b.

Well, for a, you multiply by 20/20, thus giving you 60%.

Or alternatively, you could have done 3 divided by 5, which gives you 0.

6, which we know is 60%.

Either method is perfectly fine.

For b, 7/20.

Well, we can use our knowledge on equivalent fractions by multiplying by 5/5, giving us 35%.

Alternatively, you could have used your division, 7 divided by 20, gives us 0.

35 as a decimal, which we know is 35%.

Very good if you've got either of these correct.

Let's have a look at c and d.

Well, for c and d I'm going to choose short division.

1 divided by 8 works out to be 0.

125, therefore 1/8 as a percentage is 12.

5%.

For d, 1/11 is the same as 1 divided by 11.

Working this out, it's 0.

09 recurring.

So that means 0.

09 recurring multiplied by 100 gives me 9.

09% recurring, rounded to one decimal place gives me 9.

1%.

Massive well done if you've got this one right? Great work everybody.

So now let's move on to writing any number as a percentage of another.

Well, let's have a look at this question.

What percentage of 40 is 10? Well, to do this, I'm going to use some vertical bars.

Here I have 40 next to it I'm going to identify my 10.

Now what I'm gonna do is break my 40, so I can represent that 10 a little bit easier.

So I have 30 and I have 10 still representing my 40.

So now what I'm going to do is use this to break it into equal parts.

Given the fact that I know that bottom section has to be 10 and I know the top section has to be 30, that means I can spot each section here in my vertical bar has to be 10.

So what fraction does 10 represent? Well, it's 1/4.

So 10 is 1/4 of 40.

But the question wanted be to find out what percentage of 40 is 10? Well, let's convert that quarter into a percentage.

I'm gonna multiply by 25/25 giving me 25/100, which we know is 25%.

So therefore, 10 is 25% of 40 and we've used vertical bars to visualise this.

Now what I want you to is have a little think of what percentage of 250 is 150.

We could use those vertical bars again.

So here's my 250 and here is my 150.

Let's think about drawing a vertical bar illustrating the 150 as a fraction of our 250.

Well, this has to be 250, so the top section must be 100.

I'm going to break it now into five equal parts.

So you can see each part must represent 50 in order for my total vertical bar to sum to 250.

Now I know this represents 150, which is 3/5.

So that means I know 150 is 3/5 of 250.

While using our knowledge on converting fractions to percentages, 3/5 is the same as 60 per 100, which I know is 60%.

Another way you could look at it is actually looking at what fraction of 250 is 150.

So we can represent this as a fraction, 150/250.

This can then be simplified to 3/5, and notice how we still end up with the same fraction of 3/5 and then we can convert it into a percentage.

Really well done, if you spotted that 150 is 60% of 250.

So now let's move on to a check.

You're very welcome to use these vertical bars to help or you can use your knowledge on equivalent fractions.

What percentage of 60 is 42? See if you can work it out and press pause for more time.

Great work, so let's have a look.

Well, hopefully you've spotted using a vertical bar.

This represents 42.

So that means we know the top section will have to represent 18.

I'm then going to divide it into 10 equal parts where you can see 42 is made out of 7/10 of those equal parts.

So that means 42 is 7/10 of 60.

Now we could also write it as 42/60, which cancel down to be 7/10 or we could use our vertical bars and identify 7/10.

Converting 7/10 into a percentage, we have 42 is 70% of 60.

Really done if you work this out.

And especially well done if you did it without the vertical bars.

Now we could also represent this type of problem on a double number line.

So what percentage of 50 is 35? Well, our double number line shows the percentage on the top and the amount on the bottom.

Here's my 100% and here's my 50.

So if we're asked to work out 35 as a percentage, all I've done is label the 35 as the amount and the percentage is unknown.

Now from here we know, because of that multiplicative relationship we're multiplying 50 by something to give us 35.

Working this out, we're multiplying 50 by 35/50 in order to give us our 35.

Because of that constant multiplied relationship, that means we need to do the same to 100%.

So multiplying 100% by 35/50.

Let's do a little bit of working out here.

100 times 35/50 is the same as 100 multiplied by 35, all divided by 50.

Using my knowledge on the associative law, I'm able to simplify this a touch more to be equivalent to 50/50 times 70, which is the same as 70%.

So now what we've done is we've worked out what 35 is as a percentage of 50 using our knowledge on that multiplicative relationship.

Great work, so let's have a look at a check.

I'd like you to use a double number line here to work out what percentage of 24 is 18? See if you can give it a go and press pause for more time.

Great work, so let's see how you got on.

Well, using my double number line, we know 100% is 24 and we're trying to find out what percentage is 18.

Well, to do this, I'm going to multiply my 24 by 18/24.

And because of that multiplicative relationship, I'm also gonna multiply 100 by 18/24.

Showing my working out, I know 100 multiplied by 18/24 can be written as this.

Now there are lots of different skills on multiplication of fractions here, but I've finally worked out that 100 times 18/24 is exactly the same as 75.

So therefore, I know 18 is 75% of 24.

Really well done if you've got this one right.

Now let's move on to looking at answering these questions but without representation.

And we can simply write the number as a fraction of its whole.

For example, 90 as a percentage of 270.

Let's write 90 as a percentage of 270 as a fraction, 90/270.

And I'm going to simplify here by simply identifying the highest common factor of 90 and 270, which is 90.

So that means I'm rewriting my fraction as 90 times 1 over 90 times 3.

Then simplifying, simply gives me a fraction of 1/3.

Now we use our knowledge to convert our fraction to a decimal.

I'm going to use short division again, identifying that the fraction of 1/3 is equivalent to 0.

3 recurring.

Knowing 0.

3 recurring is our decimal equivalent, we now know that 90 as a percentage of 270 is 33.

3% recurring, because we can convert our decimal into a percentage.

This was a great question and you can see how we've removed that visual representation of our double number line.

Now what I'd like you to do is a quick check question without that representation.

See if you can work out 35 as a percentage of 50.

Press pause if you need more time.

Great work, let's see how you got on.

Well, 35 as a percentage of 50 can be written as 35/50.

Simplifying, gives me 7/10.

Using short division, 7/10 is 0.

7.

So that means I know 35 as a percentage of 50 is 70%.

You may have also chosen to use equivalent fractions.

So identifying 35/50 is equivalent to 7/10.

Let's make that denominator 100.

Making the denominator 100, we have 70%.

Either method would've been absolutely fine.

Notice how we did it without visual representation.

Massive well done if you've got this one right.

So we know there are lots of ways we can write a number as a percentage of another, double number lines, vertical bars and using fractions.

Which method do you prefer? All of these are good approaches, but just remember some methods do take a little more time than others.

Now let's move on to your task.

I want you to work out the following, given your answer to two decimal places where appropriate.

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

Great work, let's move on to question 2.

Question 2 says, Laura's ill and misses her first day of school.

Now question a, wants you to work out what is her percentage attendance after one day? Now Laura's better the next day and attends school, so what is her percentage attendance after day 2? Now Laura attends the rest of that week and all of the next week.

What is her percentage attendance after day 5? d says, what is her percentage attendance after day 10? And lastly, Laura says she will need to attend every day for the rest of the year in order to get 100% attendance.

Is Laura correct? And I want you to explain why or why not? See if you can give it a go, press pause for more time.

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

There are lots of different ways that you could work this out.

So I'm just gonna give you these answers here.

Remember, rounding to two decimal places were appropriate.

Well done if you've got any of these ensuring to show you're working out.

Let's move on to question 2.

Now, question 2, what is her percentage after day 1? Well, she didn't go to school at all, so zero out of 1 is 0%.

Laura's better the next day and attends school, so what is her percentage attendance after day 2? Well, 1 out of 2 is a percentage of 50%.

Now Laura attends the rest of that week and all of the next week, what is her percentage after day 5? Well after day 5, remember she still had one day off, so that means 4 out of 5 days attending, which is 80%.

And after day 10, remember out of the 10 days, she was only present 9 of those 10 days, so that means it was a 90% attendance.

Great work if you got this one right.

And lastly, Laura says that she would need to attend every day for the rest of the year to get 100%.

Is Laura correct? Why or why not? Unfortunately, Laura's not correct.

She cannot get 100% attendance for the year as she's already missed a day.

The highest attendance that she can get to the nearest whole number is 99%, because 189 divided by 190 times 100 is on 99.

47, so on and so forth.

The lowest attendance she could get could be 5% as 9 divided by 190 times 100 is equal to 4.

7%, so on and so forth.

But let's hope Laura does attend the rest of the year in school.

Well done everybody.

So let's summarise what we've done.

We've looked at how to write any fraction as a percentage, and if the denominator is a factor of 100, we can work out the equivalent percentages easily.

We can also convert a fraction to a decimal and then convert this into a percentage for any fraction.

And there are lots of different ways we can make sense of writing one number as a percentage of another.

And in this lesson we looked at vertical bars, double number lines, and using equivalent fractions.

Great work everybody.

It was great learning with you.