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Repeating decimal to fraction

Here you will learn about converting a repeating decimal to a fraction including how to define a repeating decimal.

Students will first learn about converting repeating decimals (often called recurring decimals) to fractions as part of the number system in katex is not definedth grade.

What is converting a repeating decimal to fraction?

Converting a repeating decimal to fraction is representing the repeating decimals as a fraction without changing its value.

A repeating decimal (recurring decimal) is a decimal number that has a digit (or group of digits) right of the decimal point that repeats forever. The part that repeats can also be shown by placing a line over the first to the last digit of the repeating pattern. Repeating decimals are also known as non-terminating decimals.

For example,

katex is not defined

katex is not defined

katex is not defined

katex is not defined

are all repeating decimals.

To convert a repeating decimal to a fraction, you can form an equation. E.g.

katex is not defined

Keep track by labeling equations katex is not defined etc.

You need to be able to multiply both sides of this equation confidently by different powers of katex is not defined E.g.

katex is not defined

katex is not defined

What is converting a repeating decimal to fraction?

What is converting a repeating decimal to fraction?

Common Core State Standards

How does this relate to katex is not definedth grade math?

  • Grade 8 – The Number System (8.NS.A.1)
    Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

[FREE] Converting Fractions, Decimals and Percents Worksheet (Grade 4 to 6)

[FREE] Converting Fractions, Decimals and Percents Worksheet (Grade 4 to 6)

[FREE] Converting Fractions, Decimals and Percents Worksheet (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of converting fractions, decimals and percents. 10+ questions with answers covering a range of 4th, 5th and 6th grade converting fractions, decimals and percents topics to identify areas of strength and support!

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[FREE] Converting Fractions, Decimals and Percents Worksheet (Grade 4 to 6)

[FREE] Converting Fractions, Decimals and Percents Worksheet (Grade 4 to 6)

[FREE] Converting Fractions, Decimals and Percents Worksheet (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of converting fractions, decimals and percents. 10+ questions with answers covering a range of 4th, 5th and 6th grade converting fractions, decimals and percents topics to identify areas of strength and support!

DOWNLOAD FREE

How to convert a repeating decimal to fraction

In order to convert from a repeating decimal to a fraction, you need to:

  1. Equate the repeating decimal to a variable, katex is not defined to create katex is not defined
  2. Multiply both sides of katex is not defined by a power of katex is not defined to create katex is not defined
  3. Subtract katex is not defined from katex is not defined
  4. Divide the value by the coefficient of katex is not defined
  5. Simplify the fraction.
  6. State the answer.

Repeating decimal to fraction examples

Example 1: converting with a simple repeating decimal (one repeated digit)

Convert katex is not defined to a fraction.

  1. Equate the repeating decimal to a variable, katex is not defined to create katex is not defined

katex is not defined

2Multiply both sides of katex is not defined by a power of katex is not defined to create katex is not defined

As katex is not defined has one repeating digit, multiply by katex is not defined Remember, because you are multiplying the whole of katex is not defined by katex is not defined, you also need to multiply the variable katex is not defined by katex is not defined

katex is not defined

3Subtract katex is not defined from katex is not defined

Subtract the first equation from the second equation.

katex is not defined

4Divide the value by the coefficient of katex is not defined

katex is not defined

As katex is not defined is equal to katex is not defined and katex is not defined it can be stated that katex is not defined and katex is not defined are equal to one another.

5Simplify the fraction.

This fraction cannot be simplified as the only factor that katex is not defined and katex is not defined share is katex is not defined

6State your answer.

katex is not defined

Example 2: converting with a repeating decimal with simplifying (two repeated digits)

Convert katex is not defined to a fraction.

Equate the repeating decimal to a variable, katex is not defined to create katex is not defined

Show step

Multiply both sides of katex is not defined by a power of katex is not defined to create katex is not defined

Show step

Subtract katex is not defined from katex is not defined

Show step

Divide the value by the coefficient of katex is not defined

Show step

Simplify the fraction.

Show step

State your answer.

Show step

Example 3: converting with a repeating decimal (one repeated digit)

Convert katex is not defined to a fraction.

Equate the repeating decimal to a variable, katex is not defined to create katex is not defined

Show step

Multiply both sides of katex is not defined by a power of katex is not defined to create katex is not defined

Show step

Subtract katex is not defined from katex is not defined

Show step

Divide the value by the coefficient of katex is not defined

Show step

Simplify the fraction.

Show step

State the answer.

Show step

Example 4: converting with a repeating decimal (two repeated digits)

Convert katex is not defined to a fraction.

Equate the repeating decimal to a variable, katex is not defined to create katex is not defined

Show step

Multiply both sides of katex is not defined by a power of katex is not defined to create katex is not defined

Show step

Subtract katex is not defined from katex is not defined

Show step

Divide the value by the coefficient of katex is not defined

Show step

Simplify the fraction.

Show step

State the answer.

Show step

Example 5: converting with a repeating decimal greater than 1 (one repeated digit)

Convert katex is not defined to a fraction.

Equate the repeating decimal to a variable, katex is not defined to create katex is not defined

Show step

Multiply both sides of katex is not defined by a power of katex is not defined to create katex is not defined

Show step

Subtract katex is not defined from katex is not defined

Show step

Divide the value by the coefficient of katex is not defined

Show step

Simplify the fraction.

Show step

State the answer.

Show step

Example 6: converting with a more complex repeating decimal (two repeated digits)

Convert katex is not defined to a fraction.

Equate the repeating decimal to a variable, katex is not defined to create katex is not defined

Show step

Multiply both sides of katex is not defined by a power of katex is not defined to create katex is not defined

Show step

Subtract katex is not defined from katex is not defined

Show step

Divide the value by the coefficient of katex is not defined

Show step

Simplify the fraction.

Show step

State the answer.

Show step

Teaching tips for converting repeating decimals to fractions

  • Students should have a firm grasp on what recurring or repeating decimals are before beginning to convert them. Make sure students have a clear definition to refer back to.

  • Encourage students to identify the repeating pattern in the decimals as the repeating digits. Make sure to emphasize that the recurring part of the decimal repeats indefinitely.

  • Use of a converting repeating decimals to fractions worksheet will provide students with plenty of opportunities to practice.

  • Post step-by-step instructions within the classroom or have students write steps down in a math journal to refer back to when needed.

Easy mistakes to make

  • Multiplying by an incorrect power of katex is not defined
    You must make sure you are multiplying by the correct power of katex is not defined so the repeating part of the decimal can be eliminated. For example,
    Multiplying katex is not defined by katex is not defined does not help us eliminate the repeating decimal by subtraction. In this example you need to multiply by katex is not defined because you can now eliminate the repeating aspect of the decimal by subtraction.
    • Incorrect:
      katex is not defined
      katex is not defined does not eliminate the repeating decimal..
    • Correct:
      katex is not defined
      katex is not defined does eliminate the repeating decimal.

  • Leaving the fraction with a decimal in the numerator
    Fractions should not include a decimal. For example, katex is not defined
    In this example, multiply the numerator and denominator by katex is not defined to be left with katex is not defined

  • Not simplifying the fraction
    Always check to see if you have simplified the fraction into its simplest form.
    The easiest way to do this is by using a divisibility check to see if the numerator and denominator are divisible by another number besides katex is not defined

Practice converting a repeating decimal to fraction questions

1. What should you multiply katex is not defined by to help eliminate the repeating decimal?

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

If the decimal has one repeating digit, then multiply by katex is not defined

 

katex is not defined

2. What should you multiply katex is not defined by to help eliminate the repeating decimal?

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

If the decimal has two repeating digits, then multiply by katex is not defined

 

katex is not defined

3. What should you multiply katex is not defined by to help eliminate the repeating decimal?

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

If the decimal has two repeating digits, then multiply by katex is not defined

 

katex is not defined

4. What is katex is not defined  as a fraction?

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of katex is not defined

 

katex is not defined

 

Subtract katex is not defined from katex is not defined

 

katex is not defined

 

Divide both sides of the equation by katex is not defined

 

katex is not defined

5. What is katex is not defined  as a fraction?

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of katex is not defined

 

katex is not defined

 

Subtract katex is not defined from katex is not defined

 

katex is not defined

 

Divide both sides of the equation by katex is not defined.

 

katex is not defined

 

This does not simplify further.

6. What is katex is not defined as a fraction?

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

Set the repeating decimal equal to a variable and multiply the sides by the correct multiple of katex is not defined

 

katex is not defined

 

Subtract katex is not defined from katex is not defined

 

katex is not defined

 

Divide both sides of the equation by katex is not defined.

 

katex is not defined

 

This does not simplify further.

Repeating decimal to fraction FAQs

Is there a difference between repeating decimal numbers and recurring decimal numbers?

No, both terms can be used synonymously.

What is a surd?

A surd is an irrational number expressed in radical form. They cannot be simplified to whole numbers or fractions.

What is a terminating decimal?

Terminating decimals are decimals that contain a finite number of digits after the decimal point.

Are there decimal numbers that are neither terminating or repeating?

Yes. These are called irrational numbers and include pi katex is not defined and Euler’s number katex is not defined

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