1.5 Repeating As A Fraction

Decoding the Mystery: 1.5 Repeating as a Fraction

The seemingly simple decimal 1.5 repeating, often written as 1.5̅ or 1.555..., can be surprisingly tricky to convert into a fraction. This article will demystify this process, guiding you through various methods, explaining the underlying mathematical principles, and answering frequently asked questions. Understanding this conversion not only strengthens your grasp of decimal-fraction relationships but also enhances your mathematical reasoning skills. Let's dive in!

Understanding Repeating Decimals

Before tackling the specific case of 1.5̅, let's establish a foundational understanding of 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 (e.g., 0.3̅3̅ for one-third) or by three dots (...). The repeating part is called the repetend. Unlike terminating decimals (like 0.25 or 0.75), repeating decimals represent rational numbers – numbers that can be expressed as a fraction of two integers.

The key to converting repeating decimals to fractions lies in manipulating algebraic equations to eliminate the repeating part.

Method 1: The Algebraic Approach for 1.5̅

This method directly addresses the repeating decimal nature of 1.5̅. Let's denote the value of 1.5̅ as 'x':

• Step 1: Set up the equation: x = 1.5̅

• Step 2: Multiply to shift the repetend: Multiply both sides of the equation by 10 to shift the repeating part: 10x = 15.5̅

• Step 3: Subtract the original equation: Subtract the original equation (x = 1.5̅) from the equation in Step 2:

  10x - x = 15.5̅ - 1.5̅

• Step 4: Simplify and solve for x: This step eliminates the repeating part:

  9x = 14

  x = 14/9

Therefore, 1.5̅ is equal to the fraction 14/9.

This algebraic method provides a clear and concise pathway to convert repeating decimals into fractions. The core principle is to strategically multiply the equation to align the repeating digits, allowing for their subtraction and subsequent simplification.

Method 2: Breaking Down the Decimal (For Understanding)

While the algebraic method is efficient, let's explore a more intuitive approach to understand the conversion. We can break down 1.5̅ into its constituent parts:

1.5̅ = 1 + 0.5̅

We know that 0.5̅ represents 5/9 (this can be easily verified using the algebraic method demonstrated above, setting x = 0.5̅). Therefore:

1.5̅ = 1 + 5/9

To express this as a single fraction, we need a common denominator:

1.5̅ = 9/9 + 5/9 = 14/9

This method highlights the compositional nature of the decimal, providing an alternative perspective on the conversion process. It’s particularly useful for building an intuitive understanding of the relationship between decimals and fractions.

Further Elaboration: Converting Other Repeating Decimals

The techniques outlined above are applicable to other repeating decimals. The key adaptation is adjusting the multiplier in Step 2 of the algebraic method. For example, to convert 0.3̅, you would use a multiplier of 10; for 0.12̅, you might use 100 to shift the repeating block.

Consider the decimal 0.123̅:

1. Let x = 0.123̅
2. Multiply by 1000: 1000x = 123.123̅
3. Subtract the original equation: 999x = 123
4. Solve for x: x = 123/999

This can be further simplified by dividing the numerator and denominator by 3, resulting in 41/333.

The complexity arises when the repetend is longer. However, the core principle remains consistent: using multiplication and subtraction to isolate and eliminate the repeating digits.

The Importance of Understanding Fraction-Decimal Conversions

Converting between decimals and fractions is a fundamental skill in mathematics. It's crucial for:

• Strengthening Number Sense: Understanding the relationship between decimals and fractions strengthens your overall number sense, allowing you to more easily manipulate and understand numbers in different formats.

• Solving Real-World Problems: Many real-world applications, particularly in fields like engineering, finance, and science, require a strong understanding of fractions and their decimal equivalents.

• Improving Algebraic Skills: The methods employed to convert repeating decimals rely heavily on algebraic principles, improving your ability to manipulate and solve equations.

• Building a Solid Mathematical Foundation: Mastering this skill lays a solid foundation for more advanced mathematical concepts, including calculus and linear algebra.

Frequently Asked Questions (FAQs)

Q1: Why can't we simply truncate the repeating decimal?

Truncating a repeating decimal (e.g., rounding 1.555... to 1.555) leads to an approximation, not an exact representation. The repeating nature of the decimal implies infinite precision, which is lost upon truncation. The methods described above provide exact fractional representations.

Q2: Are all repeating decimals rational numbers?

Yes. All repeating decimals can be expressed as the ratio of two integers (a fraction), making them rational numbers. Non-repeating decimals, such as π (pi) or the square root of 2, are irrational.

Q3: What if the repeating decimal has a non-repeating part before the repeating part (e.g., 2.15̅)?

The algebraic approach still works. Treat the non-repeating part separately. For 2.15̅:

1. x = 2.15̅
2. 10x = 21.5̅
3. 100x = 215.5̅
4. Subtract 10x from 100x: 90x = 194
5. x = 194/90 = 97/45

Therefore, 2.15̅ = 97/45. You'll need to adjust the multipliers based on the length of the repeating and non-repeating sections.

Q4: Is there a shortcut for converting simple repeating decimals?

For simple repeating decimals like 0.3̅ or 0.5̅, a shortcut is to place the repeating digit(s) over as many nines as there are digits in the repetend. For example, 0.3̅ = 3/9 = 1/3, and 0.12̅ = 12/99 = 4/33. However, this shortcut doesn't apply to decimals with non-repeating parts before the repetend.

Conclusion

Converting 1.5̅ (and other repeating decimals) into a fraction involves mastering a specific algebraic technique. By strategically multiplying and subtracting equations, we can systematically eliminate the repeating digits and arrive at an accurate fractional representation. Understanding this process enhances your mathematical proficiency and number sense, providing a valuable tool for various applications. Remember, the key is to understand the underlying principles rather than simply memorizing steps. Practice with different examples to solidify your understanding and build confidence in tackling similar problems. This process not only helps you solve the specific problem of expressing 1.5̅ as a fraction but also equips you with a robust method for handling a broader range of repeating decimal conversions.