What Is 1.66666 As A Fraction

What is 1.66666... as a Fraction? Unraveling Repeating Decimals

The seemingly simple question, "What is 1.66666... as a fraction?" opens a door to a fascinating world of mathematical concepts. Understanding how to convert repeating decimals to fractions is a fundamental skill in mathematics, with applications extending far beyond the classroom. This comprehensive guide will not only answer this specific question but also equip you with the tools to tackle similar conversions effectively.

Understanding Repeating Decimals

Before diving into the conversion process, let's clarify what a repeating decimal is. A repeating decimal, also known as a recurring decimal, is a decimal number that has a digit or a group of digits that repeats infinitely. The repeating part is usually indicated by a bar placed over the repeating digits. For example:

• 0.3333... is written as 0.$\bar{3}$
• 0.142857142857... is written as 0.$\overline{142857}$

Our number, 1.66666..., has a repeating digit '6'. We can write it as 1.$\bar{6}$. This indicates that the digit 6 repeats infinitely. Understanding this notation is crucial for the conversion process.

Converting Repeating Decimals to Fractions: A Step-by-Step Guide

The method for converting repeating decimals to fractions involves algebraic manipulation. Here's a breakdown of the process, using 1.$\bar{6}$ as our example:

Step 1: Assign a Variable

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

x = 1.66666...

Step 2: Multiply to Shift the Decimal

Multiply both sides of the equation by a power of 10 that shifts the repeating digits to the left of the decimal point. Since only one digit is repeating, we multiply by 10:

10x = 16.66666...

Step 3: Subtract the Original Equation

Subtracting the original equation (Step 1) from the equation in Step 2 eliminates the repeating decimal part:

10x - x = 16.66666... - 1.66666...

This simplifies to:

9x = 15

Step 4: Solve for x

Divide both sides of the equation by 9 to solve for x:

x = 15/9

Step 5: Simplify the Fraction

The fraction 15/9 can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

x = 5/3

Therefore, 1.66666... as a fraction is 5/3.

Practical Applications and Real-World Examples

The ability to convert repeating decimals to fractions is not merely an academic exercise. It has practical applications in various fields:

• Engineering and Physics: Accurate calculations often require fractional representations for precise measurements and calculations. Repeating decimals, when converted to fractions, eliminate potential rounding errors.
• Finance and Accounting: Working with financial data often necessitates precise representation of values. Fractions provide accuracy that repeating decimals may lack.
• Computer Science: Representing numbers in binary and other number systems sometimes involves converting between decimal and fractional representations.
• Everyday Calculations: Although less obvious, the underlying principles apply to tasks like dividing quantities or sharing resources fairly.

Consider a scenario where you need to divide 5 pizzas equally among 3 people. The result is 5/3 pizzas per person, which is precisely 1.66666... pizzas. Converting the decimal to a fraction clarifies that each person gets one whole pizza and one-third of another.

Extending the Concept: More Complex Repeating Decimals

The method described above can be extended to handle more complex repeating decimals with multiple repeating digits. Let's illustrate this with an example:

Convert 0.$\overline{142857}$ to a fraction.

Step 1: Assign a Variable

x = 0.142857142857...

Step 2: Multiply to Shift the Decimal

Since six digits repeat, we multiply by 10⁶ (1,000,000):

1,000,000x = 142857.142857142857...

Step 3: Subtract the Original Equation

1,000,000x - x = 142857.142857142857... - 0.142857142857...

999,999x = 142857

Step 4: Solve for x

x = 142857/999,999

Step 5: Simplify the Fraction

This fraction can be simplified by dividing both numerator and denominator by their GCD, which is 142857:

x = 1/7

Thus, 0.$\overline{142857}$ is equivalent to 1/7.

Handling Mixed Repeating Decimals

Some decimal numbers have a non-repeating part followed by a repeating part. For example, 2.3$\bar{5}$ . The approach remains similar, but requires a slight modification:

Convert 2.3$\bar{5}$ to a fraction.

Step 1: Separate the Non-Repeating Part

Rewrite the number as the sum of its non-repeating and repeating parts: 2.3 + 0.0$\bar{5}$

Step 2: Convert the Repeating Part to a Fraction

Let y = 0.0$\bar{5}$. Following the steps from the previous examples, we get:

10y = 0.5555... 10y - y = 0.5555... - 0.0555... 9y = 0.5 y = 0.5/9 = 1/18

Step 3: Convert the Non-Repeating Part to a Fraction

2.3 = 23/10

Step 4: Add the Fractions

23/10 + 1/18 = (23 * 18 + 10)/180 = 414/180 = 23/10

Step 5: Simplify the Fraction (if needed)

The fraction 434/180 simplifies to 217/90. This gives us the same value as 2.4111...

Step 5: Combine the results: Convert each part to an improper fraction and add them. 2.3 can be written as 23/10. Then add 23/10 and 1/18 using a common denominator to obtain 217/90.

Therefore, 2.3$\bar{5}$ as a fraction is 217/90.

Conclusion: Mastering Decimal to Fraction Conversions

Converting repeating decimals to fractions is a valuable skill that bridges the gap between seemingly disparate mathematical concepts. By understanding the underlying principles and applying the step-by-step methods outlined in this guide, you can confidently tackle a wide range of problems, from simple repeating decimals to more complex mixed numbers. The practical applications of this skill extend beyond the classroom, providing a foundation for precise calculations and accurate representations of values in various real-world situations. Remember, practice is key! The more you work through these conversions, the more comfortable and proficient you will become.