Convert 0.24 To A Fraction In Simplest Terms

Converting 0.24 to a Fraction: A Comprehensive Guide

Converting decimals to fractions might seem daunting at first, but with a systematic approach, it becomes a straightforward process. This comprehensive guide will walk you through the steps of converting 0.24 to a fraction in its simplest form, explaining the underlying principles and providing you with the skills to tackle similar conversions with confidence. We'll also explore some related concepts to solidify your understanding.

Understanding Decimal and Fraction Representation

Before diving into the conversion, let's refresh our understanding of decimals and fractions. Decimals represent parts of a whole using a base-ten system. The digits to the right of the decimal point represent tenths, hundredths, thousandths, and so on. Fractions, on the other hand, represent parts of a whole using a numerator (the top number) and a denominator (the bottom number). The denominator indicates the total number of equal parts, while the numerator shows how many of those parts are being considered.

For example, 0.24 means 24 hundredths, while a fraction like 1/4 represents one out of four equal parts. Our goal is to represent the decimal 0.24 using a fraction.

Step-by-Step Conversion of 0.24 to a Fraction

The process of converting a decimal to a fraction involves several simple steps:

1. Write the decimal as a fraction with a denominator of 1:

This is the initial step and sets the foundation for further simplification. We start by writing 0.24 as a fraction:

0.24/1

2. Multiply both the numerator and the denominator by a power of 10 to remove the decimal:

Since 0.24 has two digits after the decimal point, we'll multiply both the numerator and the denominator by 10². This is equivalent to multiplying by 100.

(0.24 * 100) / (1 * 100) = 24/100

This effectively shifts the decimal point two places to the right, removing it entirely from the numerator.

3. Simplify the fraction to its lowest terms:

This is the crucial step that ensures the fraction is in its simplest form. To do this, we need to find the greatest common divisor (GCD) of both the numerator (24) and the denominator (100). The GCD is the largest number that divides both 24 and 100 without leaving a remainder.

Let's find the GCD using the prime factorization method:

• Prime factorization of 24: 2 x 2 x 2 x 3 = 2³ x 3
• Prime factorization of 100: 2 x 2 x 5 x 5 = 2² x 5²

The common factors are 2 x 2 = 4. Therefore, the GCD of 24 and 100 is 4.

Now, divide both the numerator and denominator by the GCD (4):

24/4 = 6
100/4 = 25

This gives us the simplified fraction:

6/25

Therefore, 0.24 expressed as a fraction in its simplest terms is 6/25.

Verifying the Conversion

To ensure our conversion is accurate, we can convert the fraction back to a decimal:

Divide the numerator (6) by the denominator (25):

6 ÷ 25 = 0.24

This confirms that our conversion from 0.24 to 6/25 is correct.

Understanding Greatest Common Divisor (GCD)

Finding the GCD is a crucial part of simplifying fractions. Several methods exist for finding the GCD, including:

• Prime Factorization: This method involves breaking down the numbers into their prime factors and identifying the common factors. We used this method above.

• Euclidean Algorithm: This is a more efficient method, especially for larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCD.

For example, to find the GCD of 24 and 100 using the Euclidean algorithm:

1. Divide 100 by 24: 100 = 4 x 24 + 4
2. Divide 24 by the remainder (4): 24 = 6 x 4 + 0

The last non-zero remainder is 4, confirming that the GCD of 24 and 100 is 4.

Converting Other Decimals to Fractions

The method we used for 0.24 applies to other decimals as well. The key is to:

1. Write the decimal as a fraction over 1.
2. Multiply the numerator and denominator by the appropriate power of 10 to eliminate the decimal point. The power of 10 is determined by the number of decimal places.
3. Simplify the resulting fraction to its lowest terms by finding the GCD and dividing both the numerator and denominator by it.

Let's consider an example: converting 0.375 to a fraction.

1. 0.375/1
2. (0.375 * 1000) / (1 * 1000) = 375/1000
3. Find the GCD of 375 and 1000. The prime factorization shows that the GCD is 125.
4. 375/125 = 3 1000/125 = 8
5. The simplified fraction is 3/8

Dealing with Recurring Decimals

Recurring decimals, those with repeating digits, require a slightly different approach. Let's illustrate this with the example of 0.333... (0.3 recurring).

1. Let x = 0.333...
2. Multiply both sides by 10: 10x = 3.333...
3. Subtract the first equation from the second: 10x - x = 3.333... - 0.333...
4. This simplifies to 9x = 3
5. Solve for x: x = 3/9
6. Simplify: x = 1/3

Practical Applications

The ability to convert decimals to fractions is invaluable in various fields:

• Mathematics: It's fundamental to simplifying expressions, solving equations, and working with ratios and proportions.

• Science: It's essential in scientific calculations and data analysis, where precise measurements are crucial.

• Engineering: Accurate fraction representation ensures the correct dimensions and tolerances in design and manufacturing.

• Cooking and Baking: Recipes often require precise measurements, and converting between fractions and decimals ensures accuracy.

Conclusion

Converting 0.24 to a fraction, and decimals in general, is a fundamental skill with wide-ranging applications. By understanding the steps involved, from writing the decimal as a fraction to simplifying it to its lowest terms, you can confidently tackle such conversions. The process emphasizes the importance of understanding both decimal and fraction representation and mastering techniques like finding the greatest common divisor. This skill extends beyond simple arithmetic, forming the basis for more complex mathematical and scientific applications. Remember to practice regularly to build proficiency and confidence in this essential mathematical skill.