1.3 Recurring As A Fraction

Decoding 1.3 Recurring: A Deep Dive into Converting Repeating Decimals to Fractions

Understanding how to convert repeating decimals, like 1.3 recurring (often written as 1.3̅ or 1.333...), into fractions is a fundamental skill in mathematics. This seemingly simple task underpins a deeper understanding of number systems and algebraic manipulation. This comprehensive guide will not only show you how to convert 1.3 recurring to a fraction but also why the method works, exploring the underlying mathematical principles and addressing common questions. We’ll also delve into solving similar problems, equipping you with the confidence to tackle any recurring decimal conversion.

Understanding Recurring Decimals

Before we tackle the conversion of 1.3 recurring, let's establish a clear understanding of what a recurring decimal is. A recurring decimal, also known as a repeating decimal, is a decimal number that has a digit or a group of digits that repeat infinitely. The repeating part is usually indicated by a bar above the repeating digits, as in 1.3̅. This means the digit 3 repeats indefinitely: 1.333333...

Other examples of recurring decimals include:

• 0.6666... (0.6̅)
• 0.142857142857... (0.142857̅)
• 2.718281828... (2.71828̅)

Converting 1.3 Recurring to a Fraction: The Step-by-Step Method

Now, let's focus on converting 1.3 recurring to a fraction. The core strategy involves using algebra to eliminate the repeating part of the decimal. Here's a detailed step-by-step approach:

Step 1: Assign a Variable

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

x = 1.3̅

Step 2: Multiply to Shift the Decimal Point

Multiply both sides of the equation by 10 to shift the repeating part one place to the left:

10x = 13.3̅

Step 3: Subtract the Original Equation

This is the crucial step. Subtract the original equation (x = 1.3̅) from the equation obtained in Step 2 (10x = 13.3̅):

10x - x = 13.3̅ - 1.3̅

Notice how the repeating part (.3̅) cancels out. This leaves us with:

9x = 12

Step 4: Solve for x

Divide both sides of the equation by 9 to isolate x:

x = 12/9

Step 5: Simplify the Fraction

Finally, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3:

x = 4/3

Therefore, 1.3 recurring is equal to 4/3.

The Mathematical Reasoning Behind the Method

The method described above relies on the properties of infinite geometric series. When we multiply the recurring decimal by 10, we essentially shift the decimal point, creating a new equation. Subtracting the original equation effectively isolates the repeating part, allowing us to eliminate the infinite repetition and obtain a manageable algebraic equation. This manipulation transforms the infinite decimal representation into a finite fractional representation.

Consider the decimal expansion of 1.3̅:

1.3̅ = 1 + 0.3 + 0.03 + 0.003 + ...

This is an infinite 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:

Sum = a / (1 - r) (where |r| < 1)

In our case:

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

Adding the integer part (1) gives us:

1 + 1/3 = 4/3

This confirms our earlier result obtained through the algebraic manipulation.

Tackling Other Recurring Decimals: Expanding Your Skills

The method demonstrated for 1.3̅ can be applied to other recurring decimals, although the number of steps may vary depending on the length of the repeating block.

Example 1: Converting 0.6̅ to a fraction:

1. x = 0.6̅
2. 10x = 6.6̅
3. 10x - x = 6.6̅ - 0.6̅
4. 9x = 6
5. x = 6/9 = 2/3

Example 2: Converting 0.142857̅ to a fraction:

This example involves a longer repeating block. We need to multiply by 1,000,000 (10⁶) to shift the repeating block one full cycle to the left:

1. x = 0.142857̅
2. 1000000x = 142857.142857̅
3. 1000000x - x = 142857.142857̅ - 0.142857̅
4. 999999x = 142857
5. x = 142857/999999 = 1/7 (after simplification)

Frequently Asked Questions (FAQ)

Q1: What if the recurring decimal has a non-repeating part before the repeating part?

A: Handle the non-repeating part separately. For instance, to convert 2.5̅ to a fraction:

1. Treat the non-repeating part (2) and the repeating part (0.5̅) separately.
2. Convert 0.5̅ to a fraction using the method above (resulting in 1/2).
3. Add the non-repeating part: 2 + 1/2 = 5/2

Q2: Can this method be used for all types of repeating decimals?

A: Yes, this fundamental approach applies to all recurring decimals, regardless of the length of the repeating block or the presence of a non-repeating part. However, the algebraic manipulation might require multiplying by higher powers of 10 for longer repeating blocks.

Q3: What if the recurring decimal has multiple repeating blocks?

A: This requires a slight modification to the method. You'll need to multiply by the appropriate power of 10 to align the repeating blocks before subtraction. For example, for a decimal with two repeating digits, you would multiply by 100.

Q4: Are there other ways to convert repeating decimals to fractions?

A: While the method described above is the most common and straightforward, other approaches exist, particularly involving the concept of infinite geometric series, as explained earlier. These alternative methods often require a deeper understanding of mathematical series and limits.

Conclusion: Mastering the Art of Decimal-to-Fraction Conversion

Converting recurring decimals to fractions might seem daunting initially, but with a systematic approach and a clear understanding of the underlying mathematical principles, it becomes a manageable and even enjoyable task. This guide has not only provided a step-by-step method for converting 1.3 recurring to 4/3 but also equipped you with the tools and understanding to tackle a wide range of similar problems. Remember, practice is key! The more you work through these conversions, the more confident and proficient you'll become in navigating the world of repeating decimals and fractions. By mastering this skill, you'll not only improve your mathematical abilities but also gain a deeper appreciation for the elegance and interconnectedness of mathematical concepts.