Converting Decimals to Fractions: A Comprehensive Guide

In mathematics, converting decimals to fractions is a fundamental skill that is essential for various applications. Whether you are dealing with pure or mixed recurring decimals, this guide will walk you through the process step-by-step.

Understanding Decimal Types and Conversion Methods

The process of converting a decimal number to a fraction can vary slightly based on the type of decimal. Here are some common types of decimals and their respective conversion methods:

Pure Recurring Decimals

Pure recurring decimals are those where all digits after the decimal point repeat. For example, 0.781781… can be simplified as follows:

1. Write the decimal as a fraction with a numerator and a denominator:

(frac{781 - 78}{900} frac{703}{900})

2. Simplify the fraction if possible. In this case, 703 and 900 have no common factors, so the fraction is already in its simplest form.

Mixed Recurring or Repeating Decimals

Mixed recurring decimals include both a non-repeating and a repeating part. For example, 55.781781… can be handled as follows:

1. Separate the non-repeating and repeating parts. Here, the non-repeating part is 55, and the repeating part is 781.

2. Write the decimal as a fraction, considering the repeating part:

(frac{55781 - 5578}{900} 55 frac{781 - 78}{900} 55 frac{703}{900})

3. Simplify the fraction if possible. As before, 703 and 900 have no common factors, so the fraction is already in its simplest form.

Decimals with an Integer Part

Decimals that have an integer part in addition to a repeating and non-repeating part can be converted as follows:

1. Write the integer part, the non-repeating part, and the repeating part separately.

2. Subtract the non-repeating part from the integer, and then write the resulting number along with the repeating part as a fraction:

(frac{550 - 55}{900} frac{545}{900})

3. Simplify the fraction if possible. In this case, the fraction can be further reduced as follows:

(frac{545 div 5}{900 div 5} frac{109}{180})

Step-by-Step Conversion Method

Here is a step-by-step guide to converting any decimal to a fraction:

Step 1: Identify the Decimal

Start with the decimal you want to convert. For example, let’s take 0.75.

Step 2: Count Decimal Places

Determine how many decimal places are in the number. For 0.75, there are two decimal places.

Step 3: Create the Fraction

1. Write the decimal number without the decimal point as the numerator. For 0.75, this becomes 75.

2. The denominator will be 1 followed by as many zeros as there are decimal places. For 0.75, the denominator is 100 since there are two decimal places.

So, you can write:

(frac{75}{100})

Step 4: Simplify the Fraction

Reduce the fraction to its simplest form by finding the greatest common divisor (GCD) of the numerator and denominator.

1. Find the GCD of 75 and 100. The GCD of 75 and 100 is 25.

2. Divide both the numerator and the denominator by their GCD:

(frac{75 div 25}{100 div 25} frac{3}{4})

Example: Converting 0.6 to a Fraction

Let’s convert 0.6 to a fraction:

Decimal: 0.6

Count Decimal Places: 1 decimal place.

Create the Fraction:

(frac{6}{10})

Simplify:

The GCD of 6 and 10 is 2:

(frac{6 div 2}{10 div 2} frac{3}{5})

Conclusion

So, 0.75 as a fraction is (frac{3}{4}) and 0.6 as a fraction is (frac{3}{5}).

This method can be used for any decimal, whether it is a pure recurring decimal, a mixed recurring decimal, or a decimal with an integer part. By understanding the steps and practicing with different types of decimals, you can master the art of converting decimals to fractions.