How to Convert Repeating Decimals to Fractions? Know the Steps with Examples

If you happen to be viewing the article How to Convert Repeating Decimals to Fractions? Know the Steps with Examples? on the website Math Hello Kitty, there are a couple of convenient ways for you to navigate through the content. You have the option to simply scroll down and leisurely read each section at your own pace. Alternatively, if you’re in a rush or looking for specific information, you can swiftly click on the table of contents provided. This will instantly direct you to the exact section that contains the information you need most urgently.

Transform repeating decimals into fractions like a pro with our clear and concise tutorial. Don’t let the complexity of decimals hold you back – unravel the mystery with our expert guidance.

How to Convert Repeating Decimals to Fractions?

Converting repeating decimals to fractions involves a simple pattern recognition technique. Let’s take the number 0.3333… as an example.

Steps to convert a repeating decimal to a fraction:

1. Multiply the decimal by 10: 0.3333… * 10 = 3.3333…
2. Subtract the original decimal from the product: 3.3333… – 0.3333… = 3
3. The numerator is the difference obtained in step 2, and the denominator is the number of 9s after the decimal point. In this case, the numerator is 3 and the denominator is 9.
4. Express the fraction in its lowest terms. Therefore, the fraction equivalent to 0.3333… is 3/9, which can be simplified to 1/3.

This method can be applied to any repeating decimal.

Repeating and Non-Repeating Decimals – Definition

Decimals can be classified into two main categories: repeating decimals and non-repeating decimals.

Repeating decimals, also known as recurring decimals, are decimal numbers where a specific pattern of digits repeats indefinitely after the decimal point. The repeating block of digits is called the repetend. For instance, 0.33333… (with 3 repeating indefinitely) and 0.212121… (with 21 repeating indefinitely) are examples of repeating decimals.

Non-repeating decimals, also known as terminating decimals, are decimal numbers where the digits after the decimal point do not repeat. They eventually terminate or end. For example, 0.25, 0.731, and 1.414213 are examples of non-repeating decimals.

Relationship with Rational and Irrational Numbers

Repeating decimals are always rational numbers, meaning they can be expressed as a fraction of two integers (p/q, where q is not equal to zero). Non-repeating decimals, on the other hand, can be either rational or irrational. Some non-repeating decimals, like 0.123456789…, are rational, while others, like 0.π (the decimal representation of pi) are irrational.

Conversion between Decimals and Fractions

Repeating decimals can be converted to fractions using a method called “equating like digits.” This method involves setting up two equations, each representing the repeating decimal with different variable names. Subtracting the two equations eliminates the repeating block, and the resulting equation can be solved for the variable, representing the fraction equivalent to the decimal.

Non-repeating decimals can also be converted to fractions, but the method depends on whether the decimal is terminating or non-terminating. For terminating decimals, a simple multiplication by a power of 10 can be used to eliminate the decimal point and express the number as an integer. Then, the original decimal can be equated to the new integer divided by the corresponding power of 10, resulting in a fraction. For non-terminating decimals, a similar method can be used, but the division will result in an infinite repeating sequence, which can be solved using algebraic techniques.

In conclusion, repeating and non-repeating decimals are two distinct categories of decimal numbers, with different characteristics and relationships with rational and irrational numbers. Understanding these concepts is crucial in various mathematical applications.

Repeating Decimal to Fraction Chart

Steps to Convert Repeating Decimals to Fractions

View other drafts

volume_up

Converting repeating decimals to fractions involves a straightforward process that utilizes the concept of place value and algebraic manipulation. Here’s a step-by-step guide to converting repeating decimals to fractions:

1. Identify the repeating block of digits:

   • Carefully examine the decimal number and identify the block of digits that repeats indefinitely after the decimal point. For instance, in the decimal 0.414141…, the repeating block is 41.

2. Assign a variable to the decimal:

   • Let’s represent the decimal with a variable, say ‘x’. In this case, x = 0.414141…

3. Create two equations:

   • Write two equations, one with the decimal point on the left of the repeating block and the other with the decimal point to the right of the repeating block.
   • For example, in this case: Equation 1: x = 0.414141… Equation 2: 100x = 41.4141…

4. Subtract the smaller equation from the larger equation:

   • Subtract Equation 1 from Equation 2 to eliminate the repeating decimals.
   • 100x – x = 41.4141… – 0.414141…
   • 99x = 41

5. Divide both sides by the common factor:

   • Divide both sides of the equation by the common factor, which is 99 in this case.
   • x = 41/99

6. Simplify the fraction (if possible):

   • If the fraction can be simplified, simplify it by dividing the numerator and denominator by their greatest common factor.

7. The fraction represents the equivalent decimal:

   • The simplified fraction obtained in the previous step represents the equivalent fraction of the repeating decimal.
   • Therefore, 0.414141… = 41/99.

This method can be applied to any repeating decimal, regardless of the length of the repeating block. The key is to identify the repeating block and then use algebraic manipulation to eliminate the repeating decimals.

Converting Repeating Decimals to Fractions

Converting repeating decimals to fractions involves a simple yet effective process. Here’s a step-by-step guide:

Step 1: Identify the Repeating Pattern

Start by identifying the repeating pattern in the decimal. For instance, in 0.414141…, the repeating pattern is 41.

Step 2: Set Up Two Equations

Let ‘x’ represent the decimal number. Set up two equations, one with the repeating block and the other with the entire decimal.

Equation 1: x = 0.414141…

Equation 2: 100x = 41.414141…

Step 3: Subtract the Equations

Subtract Equation 1 from Equation 2:

99x = 41

Step 4: Simplify the Fraction

Divide both sides of the equation by 99:

x = 41/99

Step 5: Express the Decimal as a Fraction

The decimal 0.414141… is equivalent to the fraction 41/99.

Some Solved Examples on Repeating Decimals to Fractions

Converting repeating decimals to fractions can be done using a simple algebraic method. Here’s a step-by-step guide with some solved examples:

Step 1: Let x represent the repeating decimal

For instance, let’s convert the repeating decimal 0.7777… to a fraction. We can represent it as:

x = 0.7777…

Step 2: Multiply both sides of the equation by 10

This will shift the decimal point one place to the right:

10x = 7.7777…

Step 3: Subtract the original equation from the new equation

This will eliminate the repeating digits:

9x = 7

Step 4: Divide both sides by 9

This will give you the fraction in its simplest form:

x = 7/9

Therefore, 0.7777… is equivalent to the fraction 7/9.

Here are some more solved examples:

Convert 0.4444… to a fraction

x = 0.4444…
10x = 4.4444…
9x = 4
x = 4/9

Convert 0.9999… to a fraction

x = 0.9999…
10x = 9.9999…
9x = 9
x = 9/9

Convert 0.12341234… to a fraction

x = 0.12341234…
100x = 12.341234…
99x = 12.217822
x = 12.217822/99

Thank you so much for taking the time to read the article titled How to Convert Repeating Decimals to Fractions? Know the Steps with Examples written by Math Hello Kitty. Your support means a lot to us! We are glad that you found this article useful. If you have any feedback or thoughts, we would love to hear from you. Don’t forget to leave a comment and review on our website to help introduce it to others. Once again, we sincerely appreciate your support and thank you for being a valued reader!

Source: Math Hello Kitty
Categories: Math