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## 8th grade (Illustrative Mathematics)

### Unit 8: Lesson 16

Lesson 15: Infinite decimal expansions- Converting repeating decimals to fractions (part 1 of 2)
- Converting repeating decimals to fractions (part 2 of 2)
- Converting repeating decimals to fractions
- Converting multi-digit repeating decimals to fractions
- Writing repeating decimals as fractions review

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# Writing repeating decimals as fractions review

CCSS.Math:

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, point, 07**

0, point, 0, start color #11accd, 7, end color #11accd is start color #11accd, 7, end color #11accd start text, start color #1fab54, h, u, n, d, r, e, d, t, h, s, end color #1fab54, end text. So, we write start color #11accd, 7, end color #11accd over start color #1fab54, 100, end color #1fab54.

## But what about repeating decimals?

Let's look at an example.

**Rewrite 0, point, start overline, 7, end overline as a simplified fraction.**

Let x equal the decimal:

Set up a second equation such that the digits after the decimal point are identical:

Subtract the two equations:

Solve for x:

Remember from the first step that x is equal to our repeating decimal, so:

*Want to learn more about writing repeating decimals as fractions? Check out this video.*

## Want to join the conversation?

- what could you do if you had 0.2 repted(13 votes)
- if it is just one integer that is repeating, such as .1111 or .2222, it is that integer over 9. .1111=1/9, etc.(13 votes)

- How do we repeating decimals to fractions? I have a problem find the repeating decimal to a fractions?(15 votes)
- 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.(5 votes)

- what does the little line at the top of the number mean(7 votes)
- 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.(11 votes)

- What is 0.78 with the 8 repeating?(2 votes)
- 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!(4 votes)

- Why is math not fun?(6 votes)
- 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.(3 votes)

- how would I simplify a repeating decimal like 0.63636363...?(5 votes)
- Let x = 0.63636363...

Two digits (63) repeat, so multiply both sides by 10^2=100 to get

100x = 63.636363...

Subtracting the first equation from the second equation accomplishes the main goal of canceling out the repeating part:

99x = 63.

Dividing both sides by 99 gives

x = 63/99 = 7/11.

So 0.63636363... converts to 7/11.(5 votes)

- hi how r u guys(8 votes)
- Well my mom kicked me out of the housevso NJ sure itvgraet(1 vote)

- How would I turn 1.83333... into a simplified fraction?(2 votes)
- The first step is to turn it into two equations with the same decimal.

So here, to get only the 3 repeating to the right of the decimal point, the first equation would be multiplied by 10, to make:

10x = 18.333...

For the second equation, multiplying it by 100 makes sense, to create a different equation. Therefore:

100x = 183.333...

Subtract the two equations from each other. Setting it up like this visually makes more sense to me, at least:

100x = 183.333...

10x = 18.333...

The decimal cancels, so it comes out to:

90x = 165

Divide both sides by 90 to make it equal to x:

x = 165/90

Simplify the fraction by dividing out 15:

x = 11/6

I hope that makes sense!(9 votes)

- How would I solve 1.83 with just the three repeating??(3 votes)
- How do we use repeating decimals and is there a better way of explaining it?(3 votes)
- Repeating decimals are useful for writing numbers like 1/3 as decimals. A way of explaining repeating decimals is that they are similar to pi (π). There is no technical end to the decimal, however unlike with irrational numbers-which we define as real numbers without being a ratio of two numbers- we know the last digit.

To answer your first question, you need to understand what a decimal really is. A decimal is used to denote rational numbers. It extends the preexisting whole numbers 1,2,3,4,5,6,7,8,9 and 0 (The Hindu-Arabic numerals which are still in use today). Most computers use binary, which only has zeros and ones. Yet, binary is insufficient for division or multiplication. It's also important to realize that software and hardware automatically store numbers like fractions as decimals.

I hope this answered your question.(4 votes)