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# Writing repeating decimals as fractions review

Review converting repeating decimals to fractions, and then try some practice problems.

## Writing decimals as fractions

To convert a decimal to a fraction, we write the decimal number as a numerator, and we write its place value as the denominator.
Example 1: $0.07$
$0.07$ is $7$ ${\text{hundredths}}$. So, we write $7$ over $100$.
$0.07=\frac{7}{100}$

## But what about repeating decimals?

Let's look at an example.
Rewrite $0.\stackrel{―}{7}$ as a simplified fraction.
Let $x$ equal the decimal:
$x=0.7777\text{…}$
Set up a second equation such that the digits after the decimal point are identical:
$\begin{array}{rl}10x& =7.7777\text{…}\\ x& =0.7777\text{…}\end{array}$
Subtract the two equations:
$9x=7$
Solve for $x$:
$x=\frac{7}{9}$
Remember from the first step that $x$ is equal to our repeating decimal, so:
$0.\stackrel{―}{7}=\frac{7}{9}$
Want to learn more about writing repeating decimals as fractions? Check out this video.

## Practice

Problem 1
Rewrite as a simplified fraction.
$0.\stackrel{―}{2}=?$

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• What is 0.78 with the 8 repeating?
• First, you want to create two equations where the repeating digit (in this case the 8) is alone on the right side of the decimal point.

So for the first equation, multiply both sides by ten, to get:

10x = 7.888...

For the second, multiply both sides by 100, to get a different equation with the same repeating eight on the right side of the decimal point:

100x = 78.888...

Then subtract the two equations. It helps to see them together:

100x = 78.888...
10x = 7.888...

The repeating 8 is subtracted out, to get:

90x = 71

Divide both sides by 90:

x = 71/90

71/90 is fully reduced, so that's the answer.

I hope that helps!
• FOR EVERYONE WHO IS STRUGGLING WITH THIS:

HERE'S HOW TO CONVERT REPEATING DECIMALS AS FRACTIONS.

Let's use 0.92222... (with the 2 repeating)

1) Since 2, the number being repeated is in the hundredth place, we move the decimal 2 places to the right. (or to the hundredth place) Doing that, we get 92.222222

2) Now the number we get from step 1 is labelled 100𝑥.
-> 100𝑥 = 92.22222

3) Let's find 10𝑥 now. This part is simple since it is simply repeating step #1 but with moving the decimal point only ONE space. So, 10𝑥 = 9.22222

4) Let's subtract 100𝑥 minus 10𝑥.
100𝑥 = 92.222
-10𝑥 = 9.222
-------------
90𝑥 = 83
= 83/90

A. Line up the numbers by the decimal point
B. Subtract 100𝑥 by 10𝑥 and 92.222 by 9.222
C. Divide the answer by 90𝑥 (put it in a fraction with 90 as the denominator)
D. That fraction is your answer! if it's an improper fraction (the numerator is larger than the denominator), you can turn it into a mixed fraction.

~~~~~~~~~~~~~~~~
I hope this helped anyone struggling with this topic! I'm not really sure if this was a good explanation but it's the best I can do :)

Don't give up if you don't get it the first time, it took me about 20 tries on the exercises to get 100%.
~~~~~~~~~
I'm willing to answer any questions so just reply to this comment and I'll try to get back to you!
• idk man im not sure if this is right
• what could you do if you had 0.2 repted
• if it is just one integer that is repeating, such as .1111 or .2222, it is that integer over 9. .1111=1/9, etc.
• How do we repeating decimals to fractions? I have a problem find the repeating decimal to a fractions?
• Converting repeating decimals to fractions
Let x equal the repeating decimal you are trying to convert to a fraction.
Examine the repeating decimal to find the repeating digit(s).
Place the repeating digit(s) to the left of the decimal point.
Place the repeating digit(s) to the right of the decimal point.
• hi what grades are yall in
• I don't understand this at all please explain
• It's okay that you don't understand it right away! Let me give you an example: write 0.6 repeating as a fraction.

To begin the process, we set up an equation. x is equal to 0.666...

x= 0.666

We are going to set up two equations, where we subtract the first equation (x= 0.666) from another equation, which we do not have yet. We want to set up the new equation so that, when we subtract our first equation (x = 0.666) from it, all numbers after the decimal place equal 0. This will give us a whole number.

To do this, we need to move our decimal to the right. In some scenarios, this may involve multiplying by numbers like 100 or 1000. However, we only need to multiply by 10 here to move the decimal one place over to the right. Our new decimal would be:

x= 6.666...

But we're not done, whatever you do to one side of the equation, you have to do to the other. So..

10x=6.666

Now we can take this new equation and subtract x=0.666 from it.

10x = 6.666
- x = 0.666

This gives us 9x=6.

We now need to "solve" for x. So, we divide by 9 to get

x= 6/9

The reason I put solve in quotation marks is because we already know what our x is! In this scenario, x equals 0.6 repeating. So

0.666 = 6/9

Finally, we simplify to get

0.666= 2/3

I always like to double check my answers, and luckily for us, these are very simple to double check. Simply punch 2 divided by 3 in your calculator, and you can verify that it equals 0.666.

I hope this helped! :))
• I got things wrong that I thought was right and I checked my work with a calculator, but Khan Academy still said I was wrong, and their answer was so confusing. 😭
• I thought it was a waste of time trying to repeat the numbers a lot, but it really helps! Just try your best and remember not to leave a fraction with a decimal or the app will go NUTTS. I learned from experience.
• what does the little line at the top of the number mean
• In a decimal number, a bar over. In a decimal number, a bar over one or more consecutive digits means that the pattern of digits under the bar repeats without end.
• Why is math not fun?
• The correct question is why is math not fun for me? (me meaning you). Math is fun for me because I have good number sense.
• i dont understand how you know what to multiply the decimal by its sometimes 10 100 1 1000 its so confusing