Understanding Recurring Decimals

Oct 13, 2025 by miodeya.com 4 views

Introduction

When dealing with recurring decimals, it's often necessary to convert them into simplified fractions. This can be achieved through various methods, including algebraic manipulation and the use of infinite geometric series. In this article, we will explore the process of converting the recurring decimal $0.828282\ldots$ into a simplified fraction.

Understanding Recurring Decimals

A recurring decimal is a decimal number that has a repeating pattern of digits. In the case of $0.828282\ldots$ , the digit 2 repeats indefinitely. To convert this decimal into a fraction, we can use the following approach:

Step 1: Define the Recurring Decimal as a Variable

Let's define the recurring decimal $0.828282\ldots$ as a variable, say $x$ . We can then express $x$ as an infinite series:

x = 0.828282\ldots

Step 2: Multiply the Decimal by a Power of 10

To eliminate the recurring decimal, we can multiply $x$ by a power of 10. Let's multiply $x$ by 100:

100x = 82.828282\ldots

Step 3: Subtract the Original Decimal from the Multiplied Decimal

By subtracting the original decimal $x$ from the multiplied decimal $100x$ , we can eliminate the recurring part:

100x - x = 82.828282\ldots - 0.828282\ldots

Step 4: Simplify the Equation

Simplifying the equation, we get:

99x = 82

Step 5: Solve for x

To solve for $x$ , we can divide both sides of the equation by 99:

x = \frac{82}{99}

Conclusion

Therefore, the recurring decimal $0.828282\ldots$ can be written as a simplified fraction: $\frac{82}{99}$ .

Simplifying the Fraction

To simplify the fraction $\frac{82}{99}$ , we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 82 and 99 is 1, so the fraction cannot be simplified further.

Final Answer

The final answer is: $\boxed{\frac{82}{99}}$

Introduction

In our previous article, we explored the process of converting the recurring decimal $0.828282\ldots$ into a simplified fraction. In this article, we will answer some frequently asked questions related to recurring decimals and simplified fractions.

Q: What is a recurring decimal?

A: A recurring decimal is a decimal number that has a repeating pattern of digits. In the case of $0.828282\ldots$ , the digit 2 repeats indefinitely.

Q: How do I convert a recurring decimal into a simplified fraction?

A: To convert a recurring decimal into a simplified fraction, you can use the following steps:

Define the recurring decimal as a variable, say $x$ .
Multiply the decimal by a power of 10.
Subtract the original decimal from the multiplied decimal.
Simplify the equation.
Solve for $x$ .

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the following methods:

List the factors of each number and find the greatest common factor.
Use the Euclidean algorithm to find the GCD.
Use a calculator or online tool to find the GCD.

Q: Can a recurring decimal be simplified further?

A: Yes, a recurring decimal can be simplified further by dividing both the numerator and the denominator by their GCD.

Q: What is the difference between a recurring decimal and a non-recurring decimal?

A: A recurring decimal is a decimal number that has a repeating pattern of digits, while a non-recurring decimal is a decimal number that does not have a repeating pattern of digits.

Q: How do I convert a non-recurring decimal into a simplified fraction?

A: To convert a non-recurring decimal into a simplified fraction, you can use the following steps:

Define the non-recurring decimal as a variable, say $x$ .
Multiply the decimal by a power of 10.
Subtract the original decimal from the multiplied decimal.
Simplify the equation.
Solve for $x$ .

Q: Can a non-recurring decimal be simplified further?

A: Yes, a non-recurring decimal can be simplified further by dividing both the numerator and the denominator by their GCD.

Q: What is the importance of converting decimals into simplified fractions?

A: Converting decimals into simplified fractions is important because it allows us to perform mathematical operations more easily and accurately.

Q: How do I apply the concept of recurring decimals and simplified fractions in real-life situations?

A: The concept of recurring decimals and simplified fractions can be applied in various real-life situations, such as:

Finance: When dealing with interest rates and investments.
Science: When dealing with measurements and calculations.
Engineering: When dealing with design and construction.

Conclusion

In conclusion, recurring decimals and simplified fractions are important concepts in mathematics that can be applied in various real-life situations. By understanding how to convert recurring decimals into simplified fractions, we can perform mathematical operations more easily and accurately.

Final Tips

Always define the decimal as a variable before converting it into a simplified fraction.
Use the correct method to convert the decimal into a simplified fraction.
Simplify the fraction further by dividing both the numerator and the denominator by their GCD.
Apply the concept of recurring decimals and simplified fractions in real-life situations.