What Is 1.3 Repeating As A Fraction

What is 1.3 Repeating as a Fraction? A Comprehensive Guide

The question, "What is 1.3 repeating as a fraction?" might seem simple at first glance. However, understanding how to convert repeating decimals into fractions reveals a fascinating interplay between decimal and fractional representation of numbers. This comprehensive guide will not only answer this specific question but will also equip you with the knowledge and techniques to tackle similar problems effectively. We'll explore the underlying mathematical principles and provide step-by-step solutions, ensuring a thorough understanding of the concept.

Understanding Repeating Decimals

Before diving into the conversion process, let's clarify what we mean by "repeating decimals." A repeating decimal is a decimal number where one or more digits repeat infinitely. These repeating digits are often indicated by a bar placed over them. For instance, 1.3 repeating, which can be written as 1.3̅ or 1.$\overline{3}$, means that the digit 3 repeats indefinitely: 1.333333...

The key to converting repeating decimals to fractions lies in recognizing the pattern of repetition and employing algebraic manipulation.

Converting 1.3 Repeating to a Fraction: A Step-by-Step Approach

To convert 1.3̅ to a fraction, we'll use a systematic approach:

Step 1: Assign a Variable

Let's represent the repeating decimal with a variable, say 'x':

x = 1.3̅

Step 2: Multiply to Shift the Repeating Part

We need to manipulate the equation so that we can subtract the repeating part. To do this, we multiply both sides of the equation by 10 (since only one digit repeats):

10x = 13.3̅

Step 3: Subtract the Original Equation

Now, subtract the original equation (x = 1.3̅) from the equation we obtained in Step 2 (10x = 13.3̅):

10x - x = 13.3̅ - 1.3̅

This simplifies to:

9x = 12

Step 4: Solve for x

Divide both sides of the equation by 9:

x = 12/9

Step 5: Simplify the Fraction

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

x = (12/3) / (9/3) = 4/3

Therefore, 1.3 repeating is equal to 4/3.

Generalizing the Method: Converting Any Repeating Decimal to a Fraction

The method we used for 1.3̅ can be generalized to convert any repeating decimal to a fraction. The key is to identify the repeating block of digits and use appropriate powers of 10 to shift the decimal point.

Let's consider a more complex example: 0.123̅

Step 1: Assign a variable:

x = 0.123̅

Step 2: Multiply to align the repeating part:

Since there are three repeating digits, we multiply by 1000:

1000x = 123.123̅

Step 3: Subtract the original equation:

1000x - x = 123.123̅ - 0.123̅

This simplifies to:

999x = 123

Step 4: Solve for x:

x = 123/999

Step 5: Simplify the fraction:

The GCD of 123 and 999 is 3, so we simplify:

x = (123/3) / (999/3) = 41/333

Dealing with Repeating Decimals with Non-Repeating Parts

Sometimes, you might encounter decimals with a non-repeating part followed by a repeating part. Let's illustrate with the example 2.13̅.

Step 1: Assign a variable:

x = 2.13̅

Step 2: Handle the Non-Repeating Part:

First, subtract the non-repeating part: 2.13̅ - 2 = 0.13̅

Step 3: Apply the previous method to the repeating part:

Let y = 0.13̅

100y = 13.13̅

100y - y = 13.13̅ - 0.13̅

99y = 13

y = 13/99

Step 4: Add back the non-repeating part:

x = 2 + y = 2 + 13/99

Step 5: Express as a single fraction:

x = (2 * 99 + 13) / 99 = (198 + 13) / 99 = 211/99

Why This Method Works: A Mathematical Explanation

The method relies on the concept of infinite geometric series. A repeating decimal can be expressed as the sum of an infinite geometric series. For instance, 0.3̅ can be written as:

0.3 + 0.03 + 0.003 + 0.0003 + ...

This is a geometric series with the first term a = 0.3 and the common ratio r = 0.1. The sum of an infinite geometric series is given by the formula:

S = a / (1 - r)

In this case:

S = 0.3 / (1 - 0.1) = 0.3 / 0.9 = 1/3

This shows the connection between the repeating decimal and the fraction. The algebraic manipulation we perform effectively calculates the sum of this infinite series.

Practical Applications of Converting Repeating Decimals to Fractions

The ability to convert repeating decimals to fractions is essential in various mathematical contexts:

• Simplifying Calculations: Fractions often lead to simpler calculations than decimals, especially when dealing with multiplication or division.
• Algebra and Calculus: Fractions are often preferred in algebraic manipulations and calculus problems.
• Computer Programming: Representing numbers as fractions can be more accurate in computer programs that deal with floating-point arithmetic.
• Engineering and Physics: Precise calculations in engineering and physics often require fractional representations.

Common Mistakes to Avoid

• Incorrectly identifying the repeating part: Ensure you correctly identify the digits that repeat indefinitely.
• Errors in algebraic manipulation: Carefully perform the multiplication and subtraction steps to avoid errors.
• Forgetting to simplify the fraction: Always simplify the resulting fraction to its lowest terms.

Conclusion: Mastering Repeating Decimals

Converting repeating decimals to fractions is a valuable skill that transcends simple arithmetic. It underpins a deeper understanding of number systems and their interrelationships. By mastering the techniques outlined in this guide, you'll not only be able to solve specific problems but also appreciate the underlying mathematical principles that govern the representation of numbers. Remember, practice is key! The more you work through examples, the more comfortable and proficient you'll become in converting repeating decimals to fractions. This skill will undoubtedly prove beneficial in various academic and professional pursuits.