What Is 0.63 Expressed As A Fraction In Simplest Form

What is 0.63 Expressed as a Fraction in Simplest Form? A Comprehensive Guide

The seemingly simple question, "What is 0.63 expressed as a fraction in simplest form?" opens a door to a deeper understanding of decimal-to-fraction conversion and fraction simplification. This comprehensive guide will not only answer this question but also equip you with the skills to tackle similar problems with confidence. We'll explore various methods, delve into the underlying mathematical principles, and offer practical examples to solidify your understanding.

Understanding Decimal Numbers and Fractions

Before we jump into the conversion, let's refresh our understanding of decimal numbers and fractions. A decimal number represents a part of a whole using a base-ten system. The digits after the decimal point represent tenths, hundredths, thousandths, and so on. A fraction, on the other hand, represents a part of a whole as a ratio of two integers – a numerator (top number) and a denominator (bottom number).

For example, 0.63 represents 63 hundredths, while a fraction like ¾ represents three-quarters. The key to converting decimals to fractions lies in recognizing the place value of the last digit.

Converting 0.63 to a Fraction

The decimal 0.63 means 63 hundredths. We can immediately write this as a fraction:

63/100

This fraction is already in a relatively simple form, but let's explore the process of simplification to ensure it's in its simplest form.

Simplifying Fractions: Finding the Greatest Common Divisor (GCD)

The process of simplifying a fraction involves reducing it to its lowest terms. This is achieved by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

To find the GCD of 63 and 100, we can use several methods:

• Listing Factors: List all the factors of 63 and 100. The largest factor common to both lists is the GCD.

  • Factors of 63: 1, 3, 7, 9, 21, 63
  • Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
  • The largest common factor is 1.

• Prime Factorization: Express both numbers as products of their prime factors. The GCD is the product of the common prime factors raised to the lowest power.

  • 63 = 3² x 7
  • 100 = 2² x 5²
  • There are no common prime factors. Therefore, the GCD is 1.

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

  100 = 1 x 63 + 37 63 = 1 x 37 + 26 37 = 1 x 26 + 11 26 = 2 x 11 + 4 11 = 2 x 4 + 3 4 = 1 x 3 + 1 3 = 3 x 1 + 0

  The last non-zero remainder is 1.

Since the GCD of 63 and 100 is 1, the fraction 63/100 is already in its simplest form.

Therefore, 0.63 expressed as a fraction in its simplest form is 63/100.

Further Exploration: Working with More Complex Decimals

Let's expand our understanding by tackling more complex decimal-to-fraction conversions.

Example 1: Converting 0.125 to a fraction

0.125 represents 125 thousandths. This can be written as 125/1000.

To simplify, we find the GCD of 125 and 1000. Using prime factorization:

• 125 = 5³
• 1000 = 2³ x 5³

The GCD is 5³. Dividing both numerator and denominator by 125, we get:

125/1000 = 1/8

Therefore, 0.125 = 1/8

Example 2: Converting 0.375 to a fraction

0.375 is 375 thousandths, or 375/1000.

Using prime factorization:

• 375 = 3 x 5³
• 1000 = 2³ x 5³

The GCD is 5³. Dividing both numerator and denominator by 125:

375/1000 = 3/8

Therefore, 0.375 = 3/8

Example 3: Converting a repeating decimal to a fraction

Repeating decimals require a slightly different approach. Let's consider 0.333... (0.3 recurring).

Let x = 0.333...

Multiplying by 10: 10x = 3.333...

Subtracting the first equation from the second:

10x - x = 3.333... - 0.333...

9x = 3

x = 3/9

Simplifying: x = 1/3

This demonstrates how to handle repeating decimals. Non-repeating decimals are generally easier to convert by writing them as a fraction over a power of 10 and simplifying.

Conclusion: Mastering Decimal-to-Fraction Conversion

Converting decimals to fractions is a fundamental skill in mathematics. Understanding the place value of decimals, finding the GCD, and applying simplification techniques are crucial steps in this process. By mastering these concepts, you can confidently transform any decimal number into its simplest fractional representation. This ability is not only important for academic success but also for practical applications in various fields, including engineering, finance, and everyday calculations. Remember to practice regularly and explore different methods to solidify your understanding and improve your efficiency. The more you practice, the easier and faster you'll become at converting decimals to fractions.