Convert 14.75 To Its Equivalent Fraction Expressed In Lowest Terms

Converting 14.75 to its Equivalent Fraction in Lowest Terms: A Comprehensive Guide

Converting decimal numbers to fractions might seem daunting at first, but with a structured approach, it becomes a straightforward process. This article will guide you through the conversion of 14.75 to its equivalent fraction in the lowest terms, explaining each step in detail and providing additional insights into fraction simplification and related concepts. We'll explore various methods and delve into the underlying mathematical principles, ensuring a comprehensive understanding for readers of all levels.

Understanding Decimal Numbers and Fractions

Before diving into the conversion, let's briefly revisit the concepts of decimal numbers and fractions.

Decimal Numbers: A decimal number is a number that uses a decimal point to separate the whole number part from the fractional part. The digits to the right of the decimal point represent tenths, hundredths, thousandths, and so on. For example, in 14.75, '14' is the whole number part, and '.75' is the fractional part representing 75 hundredths.

Fractions: A fraction represents a part of a whole. It's expressed as a ratio of two integers: the numerator (top number) and the denominator (bottom number). The numerator indicates the number of parts you have, and the denominator indicates the total number of equal parts the whole is divided into. For instance, 3/4 means you have 3 parts out of a total of 4 equal parts.

Converting 14.75 to a Fraction: Step-by-Step Guide

The conversion of 14.75 to a fraction involves several key steps:

Step 1: Express the decimal part as a fraction.

The decimal part of 14.75 is 0.75. This can be written as a fraction by considering the place value of the last digit. The last digit, 5, is in the hundredths place, meaning the denominator will be 100. Therefore, 0.75 can be written as 75/100.

Step 2: Combine the whole number and the fraction.

Now, we have the whole number 14 and the fraction 75/100. To combine them, we express the whole number as an improper fraction with the same denominator as the fractional part. This gives us:

14 = 1400/100

Adding the fractions together:

1400/100 + 75/100 = 1475/100

Step 3: Simplify the fraction to its lowest terms.

The fraction 1475/100 is not in its simplest form. To simplify it, we need to find the greatest common divisor (GCD) of the numerator (1475) and the denominator (100). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Finding the GCD can be done through several methods:

• Prime Factorization: This involves breaking down the numbers into their prime factors. The prime factors of 1475 are 5² x 59, and the prime factors of 100 are 2² x 5². The GCD is the product of the common prime factors raised to the lowest power, which is 5². Therefore, GCD(1475, 100) = 25.

• Euclidean Algorithm: This is an efficient method for finding the GCD of two numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCD.

Let's apply the Euclidean Algorithm:

1475 = 100 x 14 + 75 100 = 75 x 1 + 25 75 = 25 x 3 + 0

The last non-zero remainder is 25, confirming that the GCD is 25.

Step 4: Divide both the numerator and the denominator by the GCD.

Now, we divide both the numerator and the denominator of 1475/100 by the GCD (25):

1475 ÷ 25 = 59 100 ÷ 25 = 4

Therefore, the simplified fraction is 59/4.

Verifying the Conversion

To verify the conversion, we can convert the fraction 59/4 back to a decimal:

59 ÷ 4 = 14.75

This confirms that our conversion is correct.

Further Exploration of Fraction Simplification

Simplifying fractions is a crucial skill in mathematics. Here are some additional points to consider:

Importance of Simplifying Fractions

• Clarity: Simplified fractions are easier to understand and interpret.
• Efficiency: They make calculations simpler and faster.
• Comparison: Simplified fractions make comparing fractions easier.

Methods for Finding the GCD

Besides prime factorization and the Euclidean algorithm, other methods exist for finding the greatest common divisor, including:

• Listing Factors: List all the factors of both numbers and identify the largest common factor. This method is suitable for smaller numbers.
• Using a Calculator: Many calculators have a built-in GCD function.

Applications and Real-World Examples

Converting decimals to fractions has numerous applications in various fields, including:

• Engineering: Precision calculations often require fractions.
• Cooking and Baking: Recipes frequently use fractional measurements.
• Finance: Calculations involving interest rates and proportions often involve fractions.
• Construction: Precise measurements are essential, making fraction conversion crucial.

Conclusion

Converting 14.75 to its equivalent fraction in lowest terms results in 59/4. This process involves expressing the decimal part as a fraction, combining it with the whole number, and simplifying the resulting fraction to its lowest terms using methods like prime factorization or the Euclidean algorithm. Mastering this conversion is valuable in various mathematical and real-world applications. Understanding fraction simplification techniques enhances numerical fluency and problem-solving capabilities. Remember to always verify your conversion to ensure accuracy.