# Lesson video

In progress...

Hi, I'm Mrs. Dennett, and in today's lesson, we're going to be looking at recurring decimals where two or more digits repeat.

We're going to be changing these recurring decimals into fractions.

Before we start, we'll just recap how to solve equations involving fractions.

We have 99X = 63.

We use inverse operations to help us to find X.

As X is being multiplied by 99, we divide by 99 to get X = 63 over 99.

Be very careful with the order here.

99 is the denominator because we're dividing by 99.

This fraction can be simplified by dividing the numerator and denominator by their highest common factor, in this case 9.

So X = 7 over 9.

Here are some questions for you to try.

Pause the video to complete the task and restart the video when you've finished.

For part A, we divide both sides by 100 but be aware that is common practise to write numerators and denominators as integers.

Multiplying by 2 makes 1.

5 into 3.

So we write 3 over 200, instead of 1.

5 over 100.

Questions B and C have been simplified, dividing by 3 and 9 respectively.

We are being asked to write 0.

37 recurring as a fraction.

We use algebra to help us to do this.

We let X equal the recurring decimal.

Notice we have two digits recurring.

That's two place values.

So here we multiply by 100.

To eliminate the recurrent parts of the decimal, we take X from a 100X.

Take care in lining up the columns.

Make sure each place value is in the correct place.

We do a 100X takeaway X, which leaves us with 99X.

And you can see that the recurring part of the decimal has disappeared.

Now we just need to find what X is.

We divide by 99 to get 37 over 99.

So 0.

37 recurring, as a fraction, is 37 over 99.

Next we're going to look at the decimal with three recurring digits.

To change it to a fraction, we start by letting X equal 4.

318 recurring.

What do you think we will multiply by this time? There are three recurring digits, three place values, so we multiply by a 1000.

To eliminate the recurring part of the decimal, we now perform the subtraction.

Line up those place value columns very carefully.

We are left with 999X = 4,314.

This can now be solved.

So X is 4,314 over 999.

And this can be simplified.

So 4.

318 recurring as a fraction is equivalent to 1,438 over 333.

Here is a question for you to try.

Pause the video to complete the task and restart when you've finished.

We have two recurring digits, so we multiply by 100.

Here are some questions for you to try.

Pause the video to complete the task and restart the video when you've finished.

There's quite a variety of question here.

Remember if there were two recurring digits multiply by 100.

If there were three recurring digits multiply by 1000.

Don't be put off by the questions where the decimal is greater than one.

These fractions can also be written as mixed numbers if you want to give yourself an extra challenge.

And finally try to simplify where you can.

So for part A, 24 over 99 has been simplified to 8 over 33 by dividing the numerator and the denominator by 3.

Here's a question for you to try.

Use the skills we have just learned.

Pause the video to complete the task and restart when you've finished.

Here is the correct order.

There are different ways to answer this question.

I change 0.

38 recurring to 30 over 99 using the algebraic steps we looked at earlier.

I then noticed that I could write all the fractions with a denominator of 99.

So 1/3 becomes 33 over 99.

4/11 becomes 36 over 99.

And then I put them in order, starting with the smallest.

This is a nice opportunity to remind ourselves how to change fractions into recurring decimals.

Pause the video to have a go at this question and restart when you've finished.

7 over 11 is the same as 7 divided by 11.

We use short division to help us calculate our answer.

7 divided by 11 is 0 remainder 7.

Write down the decimal points and continue to divide.

70 divided by 11 is 6 remainder 4.

40 divided by 11 is 3 remainder 7.

70 divided by 11 is 6 remainder 4, and so on, until we spot that this is a recurring decimal.

7/11 is equivalent to 0.

63 recurring.

We placed the dots above 6 and 3 to show the first and last repeating digits.

Here is a fantastic final question for you to have a go at.

Pause the video to complete the task and restart when you've finished.

The first thing to know is that each number on the cards could have been written as 0.

9 recurring.

Did the answer to part B surprise you? This question would definitely have raised a few eyebrows in my math classroom and probably even more questions.

How can 0.

9 recurring be equivalent to one? Have to tell you that the same thing happens when you have the decimal equivalent of 1/3 and 2/3.

We have to really understand the concept of infinity as well as having some knowledge of calculus and infinite series and convergence to explain this in detail, neither of which we've got time for today, unfortunately.

But I would definitely urge you to investigate further, if you're interested.

That's all for this lesson.

Thanks for watching.