# Recurring Decimals

## Change recurring decimals into their corresponding fractions and vica versa.

##### Level 1Level 2Exam-StyleHelpMore Decimals

Do not use a calculator. You can earn a trophy if you get at least 9 questions correct and you do this activity online.

 1. This is a recurring decimal number: $$0.11111111$$... Which digit is recurring? 2. This is a recurring decimal number: $$0.43999999$$... Which digit is recurring? 3. A recurring decimal number can also be written like this: $$0.\dot 6$$. Which digit is recurring? 4. For this number: $$0.17\dot 5\dot 2$$. Which group of digits are recurring? 5. For this number: $$0.198 \dot 821 \dot 2$$. What is the period of the recurrence? 6. Convert the fraction $$\frac{1}{3}$$ to a decimal correct to 3 decimal places. 7. Convert the fraction $$\frac{2}{3}$$ to a decimal correct to 5 decimal places. 8. Convert the fraction $$\frac{1}{9}$$ to a decimal correct to 3 decimal places. 9. Convert the fraction $$\frac{6}{11}$$ to a decimal correct to 5 decimal places. 10. Write the number $$8 \frac{11}{12}$$ as a decimal correct to 7 decimal places. 11. Write the number $$7 \frac{1}{13}$$ correct to 16 significant figures. 12. What is the period of recurrence of the decimal representing the fraction $$\frac{2}{13}$$ ?
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This is Recurring Decimals level 1. You can also try:
Level 2

## Instructions

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Wednesday, August 30, 2017

What is the difference between $$0.\dot 9$$ and one?
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## Help

A decimal with a repeating digit (or set of digits) is called a recurring decimal.

For example $$0.77777777...$$ is a recurring decimal and is called "nought point seven recurring"

$$9.247347347...$$ is also a recurring decimal and is called "nine point two four seven recurring"

The period of a recurring decimal is the number of digits in the repeating section so for the second example above the period is three.

A more efficient way of writing out a recurring decimal is by only writing the repeating digit once but putting a dot over the first and last number in the repeating sequence. Another method is drawing a line over the repeating digit or digits.Here are some examples.

$$0.333333333... = 0.\dot 3 = 0.\overline 3$$

$$0.76531531531... = 0.76\dot 53\dot 1 = 0.76\overline{531}$$

A fraction can be converted to a decimal using long division; dividing the numerator by the denominator. If the decimal is recurring the repeating pattern of numbers will be spotted in the long division working. The following example shows the repeating patterns when converting $$\frac{7}{11}$$ to a decimal:

There are two common methods for converting a recurring decimal to a fraction:

### Method 1

#### 1 repeating digit

Let the recurring decimal be represented by $$x$$

$$x = 0.8888888...$$

Multiply both sides by 10 (as there is one repeating digit)

$$10x = 8.8888888...$$

Subtract the first equation from the second

$$9x = 8$$ $$x = \frac{8}{9}$$

#### 2 repeating digits

Let the recurring decimal be represented by $$x$$

$$x = 1.36363636...$$

Multiply both sides by 100 (as there are two repeating digits)

$$100x = 136.36363636...$$

Subtract the first equation from the second

$$99x = 135$$ $$x = \frac{135}{99}$$ $$x = \frac{15}{11}$$

#### 3 repeating digits

The method is the same but multiply both sides by 1000.

### Method 2

#### 1 repeating digit

Example: convert $$0.8888888...$$ to a fraction.

This method requires you to know that $$\frac19 = 0.1111111...$$

$$0.8888888...$$ is exactly eight times $$0.1111111...$$

$$\therefore 0.8888888... = \frac{8}{9}$$

#### 2 repeating digits

Example: convert $$0.45454545$$ to a fraction in its lowest terms.

This method requires you to know that $$\frac{1}{99} = 0.01010101...$$

$$0.45454545...$$ is exactly forty five times $$0.01010101...$$

$$\therefore 0.45454545... = \frac{45}{99}$$ $$0.45454545... = \frac{5}{11}$$

#### 3 repeating digits

Example: convert $$0.\dot 61\dot 2$$ to a fraction in its lowest terms.

This method requires you to know that $$\frac{1}{999} = 0.\dot 00\dot 1$$

$$0.\dot 61\dot 2$$ is exactly six hundred and twelve times $$0.\dot 00\dot 1$$

$$\therefore 0.\dot 61\dot 2 = \frac{612}{999}$$ $$0.\dot 61\dot 2 = \frac{68}{111}$$

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