Lesson video

Hi there.

My name is Ms. Lambel.

You've made a really good decision to decide to join me today to do some maths.

Come on then, let's get started.

Welcome to today's lesson.

The title of today's lesson is "Converting any recurring decimal to a fraction," and this is in the unit "Arithmetic procedures: index laws." By the end of this lesson, you'll be able to appreciate the infinite nature of recurring decimals, and you'll be able to convert between a recurring decimal and a fraction.

Some key words that we'll be using in today's lesson are terminating decimal, and recurring decimal.

A terminating decimal is one that has a finite number of digits after the decimal point, and a recurring decimal is one that has an infinite number of digits after the decimal point.

Today's lesson, we'll concentrate in the first learning cycle at writing a recurring decimal as a fraction.

We'll get really confident with that method.

And in the second learning cycle, we'll solve some more complex problems with recurring decimals.

Let's get going with that first one, writing a recurring decimal as a fraction.

Aisha and Jun are discussing writing a decimal as a fraction.

Aisha says, "We know that 0.

123 is 123/1000 when it's written as a fraction, as we can use the place value table." Jun says, "Can you write any decimal as a fraction?" Aisha's response is, "I'm not sure that all can be, but I think all terminating and recurring decimals can be.

I noticed this on my calculator.

I knew that 0.

3 recurring," remember that's what that little dot over the three means; it means it recurs, "was 1/3, and I wanted to know how many threes I needed to input before the calculator changed it into 1/3." How many threes do you need to input into your calculator before it turns 0.

333, and so on, into 1/3? Pause the video and have a go on your calculator.

How many threes do you need to input before your calculator turns 0.

333 into 1/3? Aisha says, "I had to put 17 threes before my calculator changed it to 1/3." We can use the infinite nature of recurring decimals to write any recurring decimal as a fraction.

What do you notice about the difference between the following pairs of numbers? I'm not gonna read them out, 'cause I know I'll get all tongue-tied, but I'll reveal them one at a time.

What do you notice about the difference between those pairs of numbers? Jun says, "The first one is 3, the second one is 44, the third one is 56, and the final one is 789." Did you agree with Jun? Aisha says, "We've subtracted to leave an integer." That was right, wasn't it? All of Jun's answers were integers.

Jun says, "So, it would work if you subtracted all of the recurring digits." We're now gonna have a go at writing 0.

3 recurring as a fraction, and we're gonna give our answer in its simplest form.

Now, I know we know that 0.

3 recurring as a fraction is 1/3.

It's a common fraction decimal equivalent that we know.

Also, I mentioned it already during this lesson.

But we need to learn the algebraic method for showing why 0.

3 recurring is equivalent to 1/3.

We let X equal the recurring decimal, and I've just put a few digits after decimal point, well, six.

We're going to multiply both sides of the equation by 10.

We end up with 10x equals 3.

333333.

Why do I multiply by 10? Multiplying by 10 means that when you find the difference between the two equations, you are left with an integer.

We're now gonna subtract the top equation from the bottom equation, because we're gonna find the difference just as we did earlier, because we noticed that that would mean we were left with an integer.

10x subtract X is 9x, and 3.

333333 subtract 0.

333333 is equal to 3.

We're now gonna solve this, so we need to divide both sides of the equation by nine.

We end up with X is 3/9.

The question, though, said "in its simplest form." We simplify the fraction, and 3/9 simplifies to 1/3.

Let's take a look at another one.

Write 0.

5 recurring as a fraction.

Give your answer in its simplest form.

We're going to let X equal the recurring decimal.

I'm not gonna read all the fives out 'cause I know I'm gonna get confused.

We're gonna multiply both sides of the equation by 10.

So, we end up with 10x equals.

We then subtract the top equation from the bottom equation.

10x subtract X is 9x.

And if I subtract 0.

5 recurring from 5.

5 recurring, I end up with five.

Now I divide both sides of the equation by nine 'cause we're solving to find out what X is.

So, X is equal to 5/9.

And that fraction doesn't simplify.

So, we can see now that X.

we started with X is equal to 0.

5 recurring, and we've also now got that X is 5/9.

So, therefore those two things must be equivalent.

Now I'd like you to have a go at this one.

Write 0.

6 recurring as a fraction, giving your answer in its simplest form.

Pause the video, make sure that you write down all of those steps, and come back when you're ready.

Well done.

Let's check.

So, we'll let X equal 0.

6 recurring.

We're gonna multiply both sides of the equation by 10.

We end up with 10x is 6.

6 recurring.

Subtract the top equation from the bottom equation.

10x subtract X is 9x.

And 6.

6 recurring subtract 0.

6 recurring is six.

Divide both sides by nine to solve the equation.

We end up with 6/9, and then we simplify that fraction.

We know that 6/9 is equivalent to 2/3.

Write 0.

67 recurring as a fraction, giving your answer in its simplest form.

Let X equal the recurring decimal.

We multiply both sides of the equation by 10.

We end up with 10 equals 6.

7676 recurring.

Subtract the top equation from the bottom equation.

10x subtract X is 9x.

6.

76 recurring subtract 0.

676 recurring.

"Hold on," Aisha says.

"Here the recurring parts of the decimals do not line up." No, they don't, do they? I can see that the six is with the seven, and the seven is with the six.

Jun says, "That means we will not end up with an integer.

The subtraction will still result in a recurring decimal." Can you see a way that we can align the recurring parts of the decimals? Aisha says, "If we multiply X by 100, then they would align." Jun says, "Yes, that would work." Do you agree with Aisha and Jun? Yes, they're correct.

Let X equal the recurring decimal.

We are now going to multiply both sides of the equation by 100, to end up with 100x is 67.

67 recurring.

Subtract the top from the bottom, We end up with 99x.

And then, on the right-hand side of the equation, we end up with 67.

Divide both sides of the equation by 99, and we end up with 67/99, which doesn't simplify.

Write 0.

123, and the dots this time are over the one and the three, as a fraction, giving your answer in its simplest form.

Jun says, "I'm not sure what decimal this represents." Can you write down the decimal that this represents? Write down six digits after the decimal point.

How did you get on? The dots over the one and the three show that they recur but also include any digits between them.

So, this means that the one, the two, and the three all recur.

Therefore, it represents 0.

123123.

Jun says, "I think maybe now we need to multiply by 1,000." Let X equal 0.

123123.

Do you agree with Jun? Do you think we need to multiply by 1,000 now? Jun is correct.

Three digits recur, therefore we multiply by 1000.

We let X equal our recurring decimal.

We're gonna multiply both sides of the equation by 1,000.

We're then gonna subtract the top equation from the bottom equation.

And then, we're gonna divide by 999.

And then, we're gonna simplify the fraction.

And we can write 123 as a product of 3 and 41, and 999 is a product of 3 and 333, and we know that 3/3 is 1, so it simplifies to 41/333.

For each of the following, I'd like you please to decide which power of 10 you need to multiply by to write them as fractions.

Pause the video, write down your four answers, and then when you're ready, come back.

Let's check those answers.

A would be 100.

Two digits recur.

B would be 10,000, as four digits recur.

Remember the ones between the two and the five also recur.

C would be 1,000 because three digits recur.

And D would be 10 because only one digit is recurring.

We'll have a go at this question together, and then you'll be able to have a go at the one on the right-hand side.

Write 0.

248, with the two, four, and the eight all recurring, giving your answer in its simplest form.

Let X equal the recurring decimal.

Here we're gonna multiply by 1,000 because we have three digits that are recurring.

Subtract the top equation from the bottom equation.

We end up with 999x equals 248.

Divide both sides of the equation by 999.

We end up with 248/999.

Now it's your turn.

Write 0.

347, where the three, the four, and the seven are all recurring, as a fraction in its simplest form.

Pause the video, and then come back when you're ready.

Super work.

This is what we should have.

Let X equal the recurring decimal.

Here we're gonna multiply by 1,000 because three digits recur.

1,000x equals that.

We're gonna subtract the top from the bottom, so we up with 999x equals 347.

Divide both sides of the equation by 999.

We end up with a fraction, which is 347/999.

How did you get on? Super.

Well done.

Now you are ready to have a go at this first question in the task.

Write the following as fractions.

Carefully show all of your steps.

Give your answers in their simplest form.

You can pause the video, and then when you're ready, come back.

Now let's check our answers.

1A: 4/9.

B: 7/9.

C: 6/11.

D: 205/999.

And E: 4,028/9,999.

How did you get on with those? Great work.

If you need to, you obviously rewind the video and make sure that you've got all of your steps written out correctly.

Now we can move on to more complex recurring decimals.

Here we have some recurring decimals.

Aisha says, "I've noticed something about all of these." Jun's curious, "What have you noticed?" Can you see what Aisha may be talking about? Let's see.

Let's see what Aisha is talking about.

She says, "All of the denominators are a string of nines, which is the same length as the number of digits that recur." Let's take a look at that.

The first one, one digit recurs: one 9.

The second one, two digits recur: a string of two 9s, 99.

The next one, three.

Yeah, certainly looks like that is the case, doesn't it? Jun says, "So, 0.

567 recurring is gonna be 567/999." Aisha says "Yes, but we need to carefully learn the method as these questions always say you must show all of your working." Great point there, Aisha.

Write 0.

462 recurring, giving your answer in its simplest form.

Aisha says, "See, I told you, Jun! We need to know a clear method, as my I did does not work with this one.

At least I don't think it does." Jun says, "Earlier we said that you multiply by the power of 10 that is the same as the number of digits that recur." Do you agree with Jun? Let's see if it works.

Let X equal, and here there's no dot over the four, so the four is not recurring, it's just the six and the two.

So, it's 0.

462626; I've just written six digits After decimal point.

We're gonna multiply both sides of the equation by 100 because two digits recur.

And this is what we end up with.

Subtract the top equation from the bottom equation.

Is this going to eliminate the recurring parts of the decimal? If we look here, we can see that, yes, these digits here are going to disappear.

It doesn't eliminate the entire decimal part, but we have created a terminating decimal, which we will be able to deal with.

I'm gonna subtract the top equation from the bottom equation to give us 99x equals.

what's 46.

2 subtract 0.

4? That's 45.

8.

Now we're gonna solve this equation by dividing both sides by 99, and we end up with 45.

8/99.

But we need to ensure that the numerator is an integer.

Remember, when we're writing a division, it is okay to have a decimal as part of the numerator or denominator.

But here, we're asked to give our answer as a fraction, and it's not okay to have a decimal as a numerator or a denominator.

We're going to multiply both the numerator and the denominator by 10/10.

That will mean that the numerator is then an integer.

We end up with 458/990.

Then we can simplify this to 229/495.

Let's try this one.

Let X equal the recurring decimal; or the decimal, I should say, 'cause not all of it recurs.

Multiply both sides this time by 1,000 because we can see that the five, the six, and the nine are recurring.

1,000x equals.

I'm not gonna read it out 'cause I'll get all tongue-tied.

Subtract the top equation from the bottom equation.

We end up with 999x.

The bits outside the box are going to disappear.

We get 425.

69.

Subtract a 0.

42.

It's 425.

27.

Divide both sides of the equation by 999.

And then, we need to make sure that the numerator is an integer.

So, we're gonna multiply by what here? We're gonna multiply by 100.

We end up with 42,527/99,900.

Now have a go at this check for understanding.

For each of the following, decide which power of 10 you need to multiply by to write them as fractions.

Pause the video, and when you've got your four answers, we'll check those.

Okay, let's check those.

A would be 10,000.

Four digits are recurring, the 3, 4, 5, and the 6.

B is gonna be 1000 because the five, six, and seven recur.

C would be 100 because the five and the six recur.

And D would be 10 because just the five is recurring.

We're going to write 0.

1234, where the three and the four are recurring, as a fraction in its simplest form.

Let X equal the decimal.

Here we're gonna multiply by 100 because two digits recur.

Subtract the top equation from the bottom equation.

Here we can see outside of the box, those bits are going to disappear.

So, we're left with 12.

34 subtract 0.

12, which is 12.

22.

Divide both sides of the equation by 99, and then ensure that you don't have any decimals as part of your numerator or denominator.

So, we're gonna multiply the numerator and the denominator by 100, 'cause effectively then we've multiplied by one, which so it's not changed its value.

So, we end up with 1,222/9,900, and that simplifies to 611/4,950.

Now one for you to have a go at.

Pause the video.

You can use the example on the left-hand side to help you, should you need it, and I'll be here waiting when you get back.

Well done.

Let's check.

Let X equal the decimal.

This time, three digits recurred, so we multiply by 1,000.

Subtracting gives us 999x equals 340.

5.

Divide both sides of the equation by 999.

Then we need to make sure that we have no decimals as numerators or denominators, so I multiply them by 10/10, and then simplifying.

In its simplest form, this is 227/666.

Now you are more than ready to have a go at task B.

You're gonna write the following as fractions, carefully remembering to show all of your steps.

Pause the video, and then when you've got your answers, you can come back, and I will reveal question number two.

Question number two, three, and four.

So, these questions are a bit more challenging, but you've got all of the skills to be successful at them, because you know how to write each of these as a fraction.

And then, you just use your knowledge of fractions to complete the questions.

Pause the video, and then come back when you're ready.

Let's check our answers then.

Question number 1A: the answer was 13/45.

B: 86/165.

C: 617/4,995.

And D: 997/1,980.

Question number two.

We look at the first recurring decimal, and that gives us 3/22.

The second recurring decimal is 2/9.

We know, when we're multiplying fractions together, we can multiply across the numerators and across the denominators, so we end up with 6/198, which simplifies to 1/33.

So, we've shown that 0.

136, where the three and the six recur, multiplied by 0.

2 recurring, is 1/33.

Question three.

0.

54 recurring is 6/11.

0.

5 recurring is 5/9.

Multiply those together, you get 30/99, which simplifies to 10/33.

And finally, question number four.

0.

07 recurring, where the seven recurs, is 7/90.

0.

185 recurring is 5/27.

And then, when we're dividing, remember we multiply by the reciprocal of the divisor, and we work that through.

We get 21/50.

Well done with today's lesson.

It's been quite a challenging one, I think you'll agree.

What have we looked at? We've looked at that recurring decimals can be written as fractions using an algebraic method.

We let X equal the recurring decimal; that's step one.

We then multiply by the power of 10 with the same number of digits that recur.

For example, if one digit recurs, we multiply by 10.

Two digits recur, we multiply by 10 squared, which is 100.

Three digits recur, then we multiply by 10 cubed, which is 1,000, et cetera.

We then subtract and solve the equations, giving X as a proper fraction in its simplest form.

Thank you for joining me today.

Really enjoyed working there alongside you.

I look forward to seeing you really soon.

Take care, and goodbye.