1.5 Repeating As A Fraction

Decoding the Mystery: 1.5 Repeating as a Fraction

Understanding how to convert repeating decimals, like 1.5 repeating, into fractions might seem daunting at first. But fear not! This comprehensive guide will break down the process step-by-step, revealing the underlying mathematical principles and providing you with the tools to confidently tackle similar problems. We'll explore the concept of repeating decimals, delve into the methods for converting them to fractions, and even address common misconceptions along the way. This guide is designed for everyone, from students struggling with fractions to those seeking a deeper understanding of mathematical concepts. Let's dive in!

Understanding Repeating Decimals and the Concept of Infinity

Before we tackle the conversion of 1.5 repeating to a fraction, let's define what we're working with. A repeating decimal (also known as a recurring decimal) is a decimal number that has a digit or a group of digits that repeat infinitely. These repeating digits are often indicated with a bar placed above them. For instance, 0.333... is often written as 0.3̅, where the bar signifies that the digit 3 repeats indefinitely.

The number 1.5 repeating, however, presents a slightly different scenario. It's crucial to clarify if "1.5 repeating" means 1.555... (where only the 5 repeats) or 1.555... (where the 5 repeats) or even something else. This ambiguity highlights the importance of precise notation. For the sake of clarity and to demonstrate a comprehensive approach, we will address both interpretations: 1.5̅ (meaning 1.555...) and 1.5 (meaning 1.5, a terminating decimal which is straightforwardly converted).

Case 1: Converting 1.5̅ (1.555...) to a Fraction

This is where the real challenge lies. Let's systematically convert 1.5̅ into its fractional equivalent using the following steps:

Step 1: Assign a Variable

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

x = 1.5̅

Step 2: Multiply to Shift the Decimal

We need to manipulate the equation to isolate the repeating part. Multiply both sides of the equation by 10, which shifts the decimal point one place to the right:

10x = 15.5̅

Step 3: Subtract the Original Equation

Now, subtract the original equation (x = 1.5̅) from the modified equation (10x = 15.5̅). Notice how this strategically eliminates the repeating part:

10x - x = 15.5̅ - 1.5̅

This simplifies to:

9x = 14

Step 4: Solve for x

Divide both sides by 9 to isolate x:

x = 14/9

Therefore, 1.5̅ (1.555...) expressed as a fraction is 14/9. This is an improper fraction, meaning the numerator is larger than the denominator. It can also be expressed as a mixed number: 1 5/9.

Case 2: Converting 1.5 (Terminating Decimal) to a Fraction

Converting a terminating decimal to a fraction is significantly simpler. The number 1.5 has only one digit after the decimal point.

Step 1: Write as a Fraction over 10

Write the decimal part as the numerator and 10 as the denominator because there is only one digit after the decimal.

1.5 = 15/10

Step 2: Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 15 and 10 is 5. Divide both the numerator and the denominator by 5:

15/10 = (15 ÷ 5) / (10 ÷ 5) = 3/2

Therefore, 1.5 expressed as a fraction is 3/2. This is also an improper fraction and can be expressed as a mixed number: 1 1/2.

The Mathematical Reasoning Behind the Conversion

The method we used for converting 1.5̅ to a fraction relies on the concept of infinite geometric series. A repeating decimal can be expressed as the sum of an infinite geometric series. By multiplying and subtracting equations, we effectively isolate and eliminate the infinite repeating part, leaving us with a solvable algebraic expression. This process leverages the properties of infinite geometric series to find a finite representation (the fraction) of an infinite decimal.

Addressing Common Misconceptions

A frequent source of confusion is the difference between repeating and non-repeating decimals. A repeating decimal, as we discussed, has a pattern of digits that repeat infinitely. In contrast, a terminating decimal has a finite number of digits after the decimal point.

Another common misconception involves the incorrect application of the conversion method. Remember, the key is to multiply the equation by a power of 10 that shifts the decimal point to align the repeating part, enabling its cancellation during subtraction.

Frequently Asked Questions (FAQ)

• Q: What if the repeating decimal has more than one repeating digit?

  A: The process remains the same, but you might need to multiply by a higher power of 10 (e.g., 100, 1000) to align the repeating digits before subtraction. For example, to convert 0.121212... to a fraction, you'd multiply by 100.

• Q: Can all repeating decimals be expressed as fractions?

  A: Yes, every repeating decimal can be expressed as a fraction (a rational number). This is a fundamental property of rational numbers.

• Q: How can I check if my answer is correct?

  A: You can perform long division to convert the fraction back to a decimal. If the decimal matches the original repeating decimal, your conversion is accurate. For example, dividing 14 by 9 yields 1.555..., confirming our earlier conversion.

• Q: What if the repeating part starts after some non-repeating digits?

  A: You can still apply the method, but you may need to adjust the multiplication step to align only the repeating part. Consider breaking the number into a sum of a non-repeating part and a repeating part and deal with them separately.

Conclusion

Converting repeating decimals to fractions might appear challenging at first, but with a systematic approach and an understanding of the underlying mathematical principles, it becomes a manageable and even enjoyable task. This guide has provided a detailed explanation of the conversion process for different scenarios, including the specific case of 1.5 repeating. By following the steps outlined, and practicing with various examples, you can master this essential mathematical skill. Remember, the key is understanding the concept of infinite geometric series and applying the appropriate multiplication and subtraction techniques to eliminate the repeating portion. With practice and patience, you'll become confident in handling any repeating decimal and converting it to its equivalent fraction.