Lesson video

Hi everyone.

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

We are going to be looking at "Arithmetic Procedures: Index Laws", a really interesting lesson.

I hope you enjoy it, I know I will so let's make a start.

Hi everyone, and welcome to this lesson on checking and securing understanding of converting between fractions and decimals, and it's under the unit "Arithmetic Procedures: Index Laws".

And by the end of the lesson, you'll be able to convert fractions to decimals and decimals to fractions.

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

We'll be using the word vinculum today.

And vinculum is the horizontal line placed over an expression to show that everything below that line is one group.

We'll also be looking at the words terminating decimal, and a terminating decimal is one that has a finite number of digits after the decimal point.

For example, 92.

2 is a terminating decimal, you might notice we only have one decimal place.

Another example would be 193.

3894.

This is a terminating decimal as we have four decimal places.

A non-example would be 1.

9 with a decimal dot above the 9.

That means that 9 recurs and goes on forever.

Another example would be good old pi.

Pi is not a terminating decimal.

Today's lesson will be broken into three parts.

Firstly, we'll be looking at converting terminating decimals to fractions, then we'll look at converting fractions to decimals, and then we'll be looking at common equivalents.

So let's make a start.

Well, given terminating decimals have a finite number of digits after the decimal point, we can also use a place value chart to help us identify the decimal as a fraction.

A decimal is a number that has parts that are not whole, and the place value chart splits whole numbers into 10ths, 100s, 1,000s, and so on.

For example, let's look at the terminating decimal 0.

257.

Let's convert it into a fraction.

Well, I'm going to put it in our place value chart first.

What I want you to do is have a look at the position of each of those decimal places.

Here you can see we have 7/1000, so I'm going to identify it here.

Here we have 5/100, or you could call it 51/1,000.

Here we have 2/10, so therefore 0.

257 as a fraction is 257/1,000.

And what we've successfully done is convert a decimal into a fraction using our place value charts.

What do you think the number 0.

4891 is as a fraction? See if you can use the place value chart, if it helps.

Well, let's see how you got on.

Well, first of all, here is my number 0.

4891 in my place value chart.

And looking at our place value chart, you can see the digit 1 is in the 10/1,000 column.

So that means the decimal 0.

4891 can be written as 4,891/10,000.

Now what I want you to do is have a look at the decimal 0.

32, use the place value chart that helps, and convert it into a fraction.

Let's see how you got on.

Well, here's 0.

32 using our place value chart.

You can see we can write this as a fraction as 32/100, or 32/100.

Now from here, I know I can simplify this further.

So using our knowledge on simplifying fractions, I can convert 32/100 into 8/25.

Really well done if you've got this simplified fraction of our decimal.

Now what I'm going to do is I'm going to do a quick check question with you, and then I'd like you to do a check on your own.

I want you to convert the following terminating decimal to a simplified fraction, and you can use the place value chart if it helps.

Let's have a look at 0.

78.

Well, putting in 0.

78 into our place value chart, we have it here.

I can clearly see that this is 78/100.

Then I'm going to use my knowledge on simplifying fractions to give me the simplified fraction of 39/50.

So therefore, I know 0.

78 is 39/50 as a simplified fraction.

Now what I want you to do is I want you to convert 0.

16 into a simplified fraction, 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, using our place value chart, it should be positioned here.

So that means I have 16/1,000 or 16/1,000.

I can now simplify this using our knowledge in simplifying fractions to give me 2/125.

Well done if you've got this.

So now what we're going to do is remove that place value chart.

How can you easily determine the denominator of the unsimplified fraction? For example, I know 0.

384 = 384/1,000, 0.

23 = 23/100, 0.

9 = 9/10.

These are unsimplified fractions, but how did I do that without a place value chart? Well, the number of decimal places is the same as the denominator's power of 10.

For example, here we have three decimal places so the power of 10 is 3, that's why the denominator is 1,000.

Here we have two decimal places, so the power of 10 is 2.

So that's why the denominator is 100.

Here we have one decimal place, so that means the power of 10 is 1.

So using decimal places and powers of 10 as a denominator, we can also apply this to numbers greater than 1.

For example, let's convert 3.

14 into a fraction.

First of all, I'm going to look at how many decimal places do I have.

Well, I've got two decimal places so that means it should be 314/100, because we have two decimal places so the power of 2 is 10.

Then I can use my knowledge on simplifying fractions and mixed numbers to convert this into 3 7/50.

Alternatively, we can partition the integer from the decimal and then use our skills of writing a fraction to a decimal.

For example, here we know we have the integer 3 and we have the decimal 0.

14, I'm gonna focus on the decimal for now and convert 0.

14 into a fraction, which gives me 14/100, simplified then gives me 7/50.

So that means my decimal of 3.

14 into a fraction is 3 is my inter part and the 0.

14 is 7/50, giving me 3.

14 is 3 7/50.

Now it's time for a check.

What I'd like you to do is convert 12.

35 into a mixed number.

Choose whatever method you prefer.

And part B, I want you to convert 5.

68 into a mixed number and an improper fraction.

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

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

Well, for the first part I'm gonna partition my integer from my decimal places, so I have 12 + 0.

35.

Then I'm going to look at the 0.

35 and convert it into a fraction, so 0.

35 = 7/20.

That means 12.

35 is 12 7/20.

Well done if you've got this one.

For part B, convert 5.

68 into a mixed number and an improper fraction.

I'm going to use the same process and partition my integer from my decimal.

From here, looking at my 0.

68 I'm going to convert it into a decimal, which is 17/25.

So that means 5.

68 is 5 17/25.

Converting this into an improper fraction, we have 142/25.

Great work if you got this one right.

Excellent work, everybody, so now let's have a look at your task.

For question one, I want you to convert the following into simplified fractions.

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

Fantastic, let's move on to question two.

Question two are number detectives, what you'll need to do is find out what decimal are we looking at? The first one does allow you to use a calculator.

As a decimal, I am more than 1 but less than 2.

As a simplified fraction, my denominator is 9 less than the numerator.

My denominator is a multiple of 20 and less than 150, and I have 4 non-zero decimal places.

See if you can work that one out.

For question three, you are not allowed to use your calculator here but we do know the decimal has three significant figures, and the first significant figure is 1/3 of the last significant figure.

As a mixed number, the integer is a prime and the numerator and denominator of the proper fractions are also primes.

The decimal is also less than 30 and greater than 10.

Two great questions, see if you can give it a go.

Press pause for more time.

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

Well, for question one, here are all our fractional equivalents.

Remember, don't forget to simplify.

Press pause if you need more time to mark.

Well done.

For question two, did you get 1.

125, which as a fraction is 89/80? Massive well done if you got this one.

And for question three there are two answers; 23.

6 which is 23 3/5, and 29.

6 which is 29 3/5.

If you've got one of those, well done.

If you have both of them, massive well done as that was tough.

Great work, everybody.

So let's have a look at the second part of our lesson, converting fractions to decimals.

Well, a place value chart is very useful when converting a fraction to a decimal.

For example, let's look at the fraction 3/10.

What do you think that is as a decimal? Well, it has 3/10 so that means I simply put the digit 3 in the 10ths column, so 3/10 of the decimal is 0.

3.

What do you think the fraction 231/1,000 is as a decimal? Use the place value chart if it helps.

Well, we know we have 1,000th, we know we have 3/100 and we know we have 2 1/10s, so that means 0.

231 is equivalent to 231/1,000.

Well done if you've got this.

But sometimes the denominator is not given as a power of 10, so we need to use our knowledge on equivalent fractions to convert the denominator to a power of 10.

For example, 3/5.

We can convert this denominator into a power of 10.

I'm going to multiply 3/5 by 2/2.

Remember, that's equivalent to 1.

Using our knowledge on multiplication of fractions, this is the same as 6/10, and 6/10 can be easily calculated or seen using a place value chart, so it gives me the decimal of 0.

6.

Let's have a look at another example, 37/25.

What power of 10 do you think we should use for that denominator? Well, 100 would be a good choice so multiplying 37/25 by 4/4, using our knowledge on multiplying fractions, we have 148/100 which then converts to 1.

48.

Another example would be 3/8.

Let's think about that denominator.

What power of 10 could we make that denominator? Well, we could make it 1,000.

So multiplying by 125/125, remember that's equivalent to 1, we have a denominator of 1,000 which gives me the equivalent fraction of 375/1,000, which I know is 0.

375.

So converting the denominator to our power of 10 makes the conversion of our fraction to a decimal a little bit easier.

What I want you to do is I'd like you to do a check.

I'm going to do the first one, and I'd like you to work out the second one.

Work out the decimal value of the fraction 13/5.

Well, for me, I'm going to look at that denominator and think I can make that denominator 10 by multiplying by 2/2.

Then using my knowledge on multiplying fractions, this is the same as 13 x 2 over 5 x 2, giving me the equivalent fraction of 26/10.

From here, I can simply divide, giving me the decimal value of 2.

6.

13/5 is 2.

6 as a decimal.

Now I'd like you to try a check.

I want you to work out the decimal value of the fraction 31/25.

So you can give it a go, press pause for more time.

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

Well, I'm going to use the denominator of 100, so multiplying my fraction by 4/4 gives me an equivalent fraction of 124/100.

Dividing then gives me the answer of 1.

24 is the decimal equivalent of 31/25.

Massive well done if you got this one right.

So now I want you to do another check question, and I don't want you to use a calculator.

I want you to calculate the decimal value of the following.

So you can give it a go, press pause for more time.

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

Well, for each of these, you should have had these decimal equivalents.

Massive well done if you got this one right.

Okay, sometimes the denominator is not a multiple of 10, so therefore we can use short division given the vinculum is the same as a division.

For example, let's write 1/3 as a decimal.

Well, 1/3 means 1 / 3, and we can use short division to calculate this.

So how many 3s go into 1? 0.

Then we put our decimal place, then we do our trailing 0s.

How many 3s go into 10? Well, it's 3.

Don't forget our trailing 0.

And what was remaining? Well, it was 1.

How many 3s go into 10? Well, it's 3.

Another trailing 0.

And what was remaining? It was 1, and so on and so forth.

So therefore, we know 1/3 as a decimal is 0.

3 recurring.

Remember, that little dot indicates the digit 3 is recurring.

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

I'll be doing the first one, and I'd would like you to do the second one.

Using short division, I want us to work out the decimal of this fraction, and we also need to write our answer using the correct notation.

Well, 1/12 is the same as 1 / 12.

Using short division, how many 12s go into 1? 0.

Then we put our decimal point and our trailing 0.

We still haven't dealt with that 1.

How many 12s go into 10? Well it's 0 again, so we put our trailing 0 and that 10 which we didn't deal with before, so how many 12s go into 100? It's 8, remaining 4, and then we have another trailing 0.

How many 12s go into 40? Well, it's 3.

What's remaining? 4, and we have another trailing 0.

Then how many 12s go into 40? Well it's 3, so on and so forth.

So that means 1/12 as a decimal is 0.

083, and the dot is above the 3 as the 3 is the only digit that recurs.

Now it's time for your question.

What I want you to do is use short division to work out the decimal of this fraction.

And make sure you do write your answer using the correct notation.

In other words, if it recurs, use that dot correctly.

So if you can give it a go, press pause for more time.

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

Well, this means 1 / 11, and using my short division you should have some sort of working out that looks like this.

And you would've spotted that we have 09 is recurring, so one 11th = 0.

09 with a dot above the 0 and the dot above the 9, as these two digits recur.

Really well done if you got this.

Well done.

So let's have a look at another check, but the numerator is not 1.

Still using short division, I want you to work out the decimal equivalent of the following fraction and give your answer to three decimal places.

So you can give it a go, press pause for more time.

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

Well, 3/7 means 3 / 7.

So using my short division, I have this wonderful working out.

Now, given the fact that the question wanted me to give my answer to three decimal places, I needed to work out four decimal places so I know how to round correctly.

So that means, rounding 3/7 to three decimal places gives me 0.

429.

Massive well done if you got this.

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

I want you to convert the following fractions into decimals, and I want you to round three decimal places where appropriate.

So you can give it a go, press pause for more time.

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

Here we have some fractions.

Put the following fractions in ascending order, smallest to largest.

You have a couple of different options here, I think it might be slightly easier to convert them into a decimal.

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

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

So you should have had these decimal equivalents.

Massive well done if you've got these answers.

Press pause if you need more time to mark.

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

I've decided to put them into decimals just because it's a little bit easier.

Here are the decimal equivalents, and it's so much easier for me to spot the ascending order.

I should have 211/500, 9/20, 113/250, and then 23/50.

Well done if you got this.

And for part B, here are my decimal equivalents.

So from here, I can order now, giving me in ascending order 17/21, 81/100, 9/11 and 8/9.

Well done if you got these.

Excellent work everybody, so now let's have a look at common equivalents.

Now, there are certain facts that you just need to know in life.

For example, can you recall the answers to the following? What's the capital city of France? What quantity is a dozen? What common element does H2O represent? And how many continents are there? And I want you to name them.

These are common facts, I'll give you a couple of seconds.

Well, let's see how you got on.

Well, for A, I'm hoping you know the capital city of France is Paris.

A dozen is represented as 12, H2O is water.

How many continents are there? There are seven continents, Asia, Africa, North America, South America, Antarctica, Europe, and Oceania.

Hopefully, you knew these common facts and, if not, you know them now.

So just like common facts, you should learn common equivalents.

Do you think you can fill these decimal equivalents? Have a little think.

Well, I'm hoping you know that a half is 0.

5, 1/3 is 0.

3 recurring, 1/4 is not 0.

25, 1/5 is 0.

2, 1/8 is 0.

125, and 1/10 is 0.

1.

Just like your common facts that I hope you knew, you've got to learn these common equivalents because if you know them, they save you so much time later on.

And knowing these common equivalents allows you to calculate with them too.

For example, what's the decimal equivalent to 2/3? Well, we know 2/3 is 1/3 x 2.

We should know 1/3 is 0.

3 recurring, so we multiply that by 2, thus giving us 0.

6 recurring is 2/3.

So knowing the equivalent of 1/3 helps us quickly calculate the decimal equivalent of 2/3.

Now what I want you to do is I want you to use that common equivalent knowledge and identify the decimal equivalent of the following.

So if you can give it a go, press pause for more time.

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

Well, I'm hoping you spotted 3/4 is three lots of 1/4.

If we know 1/4 is not 0.

25, times it by 3, we've got 0.

75.

4/5? Well, that's 1/5 x 4.

So it's 0.

2 x 4, which is 0.

8.

7/10? Well, that's 1/10 x 7, so it's 0.

1 x 7, which is 0.

7.

1/8? Well, that's 1/8 x 3.

So we know 1/8 is 0.

125 x 3 gives me 0.

375.

Well done, and if you use those common equivalents it's more efficient to calculate those decimal equivalents.

We can use these common equivalents to help us order too.

For example, put 0.

89, 9/10, 7/8 and 84% in order.

Well, to do this I'm choosing to convert them into decimals first, and I'm going to use as the common known equivalence.

0.

89, well, that's 0.

89.

9/10, converting that to a decimal is 0.

9.

7/8, converting to a decimal is 0.

875.

And 84%, converting to a decimal is 0.

84.

Sometimes it does help to make them all have the same number of decimal places.

This is preference, but I do know it helps some students.

So all I'm doing now is making them all have three decimal places.

And then from here I can clearly see the ascending order.

So which is the smallest? Well, the smallest is 84%, 0.

84.

Then after that it's 7/8, 0.

875.

After that it's my 0.

89, and then finally my 9/10.

Remember to order what was given to you in the question.

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

I want you to put the following in ascending order.

So you can give it a go, press pause for more time.

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

Well, converting them all into decimals I have 0.

27, 0.

25, 0.

26, 0.

2, 0.

255, and 0.

375.

Same again, I know it does help some students to have them all with the same number of decimal places.

Then from here we can identify the ascending order quickly, 1/5, 1/4, 20.

255, 26%, 0.

27 and, finally, 3/8.

Remember to order what was given to you in the question.

Well done, so now it's time for your task.

Without any working out, I want you to write the decimal equivalents to these followings.

Give it a go, press pause if you need more time.

Great work.

Question two, without any working out, write the decimal equivalents to these.

Press pause if you need more time.

Fantastic.

For question three, I want you to put the following in ascending order.

That last question was a bit hard so I put a little hint.

Remember, half of 1/3 is 1/6, and half of 1/10 is 1/20.

So you can give this a go, press pause for more time.

Fantastic work everybody, let's have a look at a really tough question.

Question four wants you to use the digits below, 1, 2, 3, 4, 4, 5, 10, 10, 17, and 21, to create five fractions which are in ascending order and, without any working out, show the decimal equivalents.

This is a really tough, meaty question, so well done if you can give this a go.

Press pause for more time.

Well done, so let's look at these answers.

We should have these decimal equivalents.

Well done if you got this one right.

For question two, using those common equivalents you should have these decimals.

Really well done if you got this one right.

For question three, let's put them in ascending order and you have this.

Press pause if you need more time to mark.

Well done.

And finally, for question four, this is one example.

Very well done if you got this example or another one, and the decimal equivalents are 0.

25, 0.

75, 1.

25, 1.

7, and 2.

1.

Very well done if you got this.

Great work, everybody.

So in summary, finding the fractional equivalent to a terminating decimal can be done using a place value chart.

Converting fractions to decimals can be simply done when the denominator is a factor of a power of 10.

However, where this is not possible, we can use short division.

Finally, knowing the common equivalents is very important and allows greater efficiency when using fractions, decimals, and/or percentages.

Great work everybody, it was wonderful learning with you.