Convert 0.27 To A Rational Number In Simplest Form

Converting 0.27 to a Rational Number in Simplest Form: A Comprehensive Guide

Converting decimal numbers to rational numbers (fractions) is a fundamental concept in mathematics. This guide provides a comprehensive walkthrough of how to convert the decimal 0.27 into its simplest rational form, explaining the process step-by-step and exploring related concepts. We will delve into the definition of rational numbers, discuss different approaches to the conversion, and finally, provide additional examples to solidify your understanding.

Understanding Rational Numbers

Before we begin the conversion, it's essential to understand what a rational number is. A rational number is any number that can be expressed as a fraction p/q, where p and q are integers, and q is not equal to zero. Essentially, it's a number that can be written as a ratio of two whole numbers. Decimals that terminate (end) or repeat are always rational numbers. Decimals that neither terminate nor repeat are irrational numbers (like pi or the square root of 2).

Method 1: Using the Place Value System

The most straightforward method for converting a terminating decimal like 0.27 into a fraction uses the place value system.

Step 1: Identify the Place Value of the Last Digit

In the decimal 0.27, the last digit (7) is in the hundredths place.

Step 2: Write the Decimal as a Fraction

Since the last digit is in the hundredths place, we can write the decimal as a fraction with a denominator of 100:

0.27 = 27/100

Step 3: Simplify the Fraction

Now, we need to simplify the fraction to its simplest form. To do this, we find the greatest common divisor (GCD) of the numerator (27) and the denominator (100). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

In this case, the GCD of 27 and 100 is 1. Since the GCD is 1, the fraction is already in its simplest form.

Therefore, 0.27 as a rational number in simplest form is 27/100.

Method 2: Using the Algebraic Approach

This method is particularly helpful when dealing with repeating decimals, but it also works for terminating decimals.

Step 1: Set the Decimal Equal to x

Let x = 0.27

Step 2: Multiply by a Power of 10

We multiply both sides of the equation by 100 (since there are two digits after the decimal point):

100x = 27

Step 3: Subtract the Original Equation

Subtract the original equation (x = 0.27) from the equation obtained in step 2:

100x - x = 27 - 0.27

99x = 26.73 (We can remove the decimal by multiplying by 100)

9900x = 2673

Step 4: Solve for x

Divide both sides by 9900:

x = 2673/9900

Step 5: Simplify the Fraction

Now, we simplify the fraction by finding the GCD of 2673 and 9900. The GCD is 9. Therefore we divide both the numerator and the denominator by 9:

x = (2673/9) / (9900/9) = 297/1100

This approach seems to yield a different result, but this is due to an error in the calculations. Let's correct the method. The correct approach uses the fact that the decimal terminates.

Let x = 0.27

Multiplying by 100:

100x = 27

Solving for x:

x = 27/100

This method confirms that the simplest form is 27/100. The previous algebraic method was unnecessarily complicated for a terminating decimal. It's best suited for repeating decimals.

Why the First Method is Preferred for Terminating Decimals

For terminating decimals, the place value method is significantly more efficient and less prone to errors. The algebraic method, while versatile, introduces unnecessary complexity for simple cases like 0.27. Choosing the right method depends on the type of decimal you're working with.

Further Examples

Let's consider a few more examples to illustrate the conversion process:

Example 1: Converting 0.75 to a rational number

1. Place Value: The last digit is in the hundredths place, so we write it as 75/100.
2. Simplification: The GCD of 75 and 100 is 25. Dividing both by 25, we get 3/4.

Therefore, 0.75 = 3/4

Example 2: Converting 0.125 to a rational number

1. Place Value: The last digit is in the thousandths place, so we write it as 125/1000.
2. Simplification: The GCD of 125 and 1000 is 125. Dividing both by 125, we get 1/8.

Therefore, 0.125 = 1/8

Example 3: Converting 0.6 to a rational number

1. Place Value: The last digit is in the tenths place, so we write it as 6/10.
2. Simplification: The GCD of 6 and 10 is 2. Dividing both by 2, we get 3/5.

Therefore, 0.6 = 3/5

Example 4: A repeating decimal (0.333...)

Repeating decimals require the algebraic approach. Here's how:

Let x = 0.333...

10x = 3.333...

Subtracting the first equation from the second:

9x = 3

x = 3/9 = 1/3

Conclusion

Converting decimals to rational numbers is a crucial skill in mathematics. For terminating decimals, the place value method offers a simple and efficient approach. However, understanding the algebraic method is beneficial for handling repeating decimals. Remember to always simplify the resulting fraction to its lowest terms by finding the greatest common divisor of the numerator and denominator. Mastering this conversion allows you to work comfortably with different number systems and solve a wide range of mathematical problems. Practice these methods with various decimal numbers to enhance your understanding and speed. This will greatly improve your problem-solving capabilities in mathematics and related fields. The examples provided offer a strong foundation for tackling more complex conversions. Remember to always double-check your simplification to ensure you have reached the simplest form.