Converting Repeating Decimal Fractions

Oct 13, 2025 by pynkocean.com 84 views

Introduction

In mathematics, converting decimal fractions to their simplest form is an essential skill that helps us express numbers in a more compact and meaningful way. This article will guide you through the process of converting decimal fractions to the form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b$ is not equal to zero. We will also explore the concept of repeating decimals and how to convert them to their simplest form.

Converting Non-Repeating Decimal Fractions

A non-repeating decimal fraction is a decimal that has a finite number of digits after the decimal point. To convert a non-repeating decimal fraction to its simplest form, we can use the following steps:

Step 1: Move the Decimal Point

To convert a non-repeating decimal fraction to its simplest form, we can start by moving the decimal point to the right until we have a whole number. For example, let's consider the decimal fraction 0.425.

Step 2: Multiply by a Power of 10

Once we have moved the decimal point, we can multiply the decimal fraction by a power of 10 to eliminate the decimal point. In this case, we can multiply 0.425 by 1000 to get 425.

Step 3: Express as a Fraction

Now that we have a whole number, we can express it as a fraction by dividing it by the power of 10 that we multiplied it by. In this case, we can express 425 as a fraction by dividing it by 1000.

Step 4: Simplify the Fraction

Finally, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 425 and 1000 is 25, so we can simplify the fraction by dividing both numbers by 25.

Example 1: Converting 0.425 to Simplest Form

Let's apply the steps above to convert 0.425 to its simplest form.

Move the decimal point: 0.425 → 425
Multiply by a power of 10: 425 × 1000 = 425000
Express as a fraction: 425000 ÷ 1000 = 425/1
Simplify the fraction: 425 ÷ 25 = 17

Therefore, the simplest form of 0.425 is 17/1.

Converting Repeating Decimal Fractions

A repeating decimal fraction is a decimal that has a repeating pattern of digits after the decimal point. To convert a repeating decimal fraction to its simplest form, we can use the following steps:

Step 1: Identify the Repeating Pattern

To convert a repeating decimal fraction to its simplest form, we need to identify the repeating pattern of digits after the decimal point. For example, let's consider the decimal fraction 1.51\dot{3}.

Step 2: Multiply by a Power of 10

Once we have identified the repeating pattern, we can multiply the decimal fraction by a power of 10 to eliminate the decimal point. In this case, we can multiply 1.51\dot{3} by 10 to get 15.13\dot{3}.

Step 3: Subtract the Original Decimal Fraction

Now that we have multiplied the decimal fraction by a power of 10, we can subtract the original decimal fraction from the result to eliminate the repeating pattern. In this case, we can subtract 1.51\dot{3} from 15.13\dot{3} to get 13.62.

Step 4: Express as a Fraction

Now that we have eliminated the repeating pattern, we can express the result as a fraction by dividing it by the power of 10 that we multiplied the original decimal fraction by. In this case, we can express 13.62 as a fraction by dividing it by 10.

Step 5: Simplify the Fraction

Finally, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 1362 and 100 is 2, so we can simplify the fraction by dividing both numbers by 2.

Example 2: Converting 1.51\dot{3} to Simplest Form

Let's apply the steps above to convert 1.51\dot{3} to its simplest form.

Identify the repeating pattern: 1.51\dot{3}
Multiply by a power of 10: 1.51\dot{3} × 10 = 15.13\dot{3}
Subtract the original decimal fraction: 15.13\dot{3} - 1.51\dot{3} = 13.62
Express as a fraction: 13.62 ÷ 10 = 1362/100
Simplify the fraction: 1362 ÷ 2 = 681, 100 ÷ 2 = 50

Therefore, the simplest form of 1.51\dot{3} is 681/50.

Conclusion

Q: What is a decimal fraction?

A: A decimal fraction is a number that has a decimal point and a finite or infinite number of digits after the decimal point.

Q: What is the simplest form of a decimal fraction?

A: The simplest form of a decimal fraction is a fraction in the form $\frac{a}{b}$ , where $a$ and $b$ are integers and $b$ is not equal to zero.

Q: How do I convert a decimal fraction to its simplest form?

A: To convert a decimal fraction to its simplest form, you can follow these steps:

Move the decimal point to the right until you have a whole number.
Multiply the decimal fraction by a power of 10 to eliminate the decimal point.
Express the result as a fraction by dividing it by the power of 10 that you multiplied the original decimal fraction by.
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Q: What is a non-repeating decimal fraction?

A: A non-repeating decimal fraction is a decimal fraction that has a finite number of digits after the decimal point.

Q: What is a repeating decimal fraction?

A: A repeating decimal fraction is a decimal fraction that has an infinite number of digits after the decimal point, with a repeating pattern of digits.

Q: How do I convert a repeating decimal fraction to its simplest form?

A: To convert a repeating decimal fraction to its simplest form, you can follow these steps:

Identify the repeating pattern of digits after the decimal point.
Multiply the decimal fraction by a power of 10 to eliminate the decimal point.
Subtract the original decimal fraction from the result to eliminate the repeating pattern.
Express the result as a fraction by dividing it by the power of 10 that you multiplied the original decimal fraction by.
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

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

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 largest common factor.
Use the Euclidean algorithm to find the GCD.
Use a calculator or online tool to find the GCD.

Q: What is the difference between a decimal fraction and a fraction?

A: A decimal fraction is a number that has a decimal point and a finite or infinite number of digits after the decimal point, while a fraction is a number that is expressed as a ratio of two integers.

Q: Can I convert a decimal fraction to a fraction?

A: Yes, you can convert a decimal fraction to a fraction by following the steps outlined above.

Q: Can I convert a fraction to a decimal fraction?

A: Yes, you can convert a fraction to a decimal fraction by dividing the numerator by the denominator.

Q: What are some common decimal fractions?

A: Some common decimal fractions include:

0.5
0.25
0.75
0.125
0.375

Q: How do I convert a decimal fraction to a percentage?

A: To convert a decimal fraction to a percentage, you can multiply the decimal fraction by 100.

Q: How do I convert a percentage to a decimal fraction?

A: To convert a percentage to a decimal fraction, you can divide the percentage by 100.

Q: What are some real-world applications of decimal fractions?

A: Decimal fractions have many real-world applications, including:

Finance: Decimal fractions are used to express interest rates, investment returns, and other financial calculations.
Science: Decimal fractions are used to express measurements, such as temperature, pressure, and density.
Engineering: Decimal fractions are used to express dimensions, such as length, width, and height.
Cooking: Decimal fractions are used to express recipes, such as ingredient ratios and cooking times.