Convert 71 2/3 To A Fraction In Lowest Terms

Converting 71 2/3 to a Fraction in Lowest Terms: A Comprehensive Guide

Converting mixed numbers to improper fractions is a fundamental skill in mathematics, crucial for various applications from basic arithmetic to advanced calculus. This comprehensive guide will walk you through the process of converting the mixed number 71 2/3 into an improper fraction in its lowest terms, explaining each step in detail and providing additional examples for a thorough understanding. We'll also explore the underlying concepts and their practical significance.

Understanding Mixed Numbers and Improper Fractions

Before diving into the conversion, let's clarify the terms involved.

• Mixed Number: A mixed number combines a whole number and a proper fraction. A proper fraction has a numerator (top number) smaller than its denominator (bottom number). Our example, 71 2/3, is a mixed number: 71 is the whole number, and 2/3 is the proper fraction.

• Improper Fraction: An improper fraction has a numerator greater than or equal to its denominator. Improper fractions represent values greater than or equal to one. Our goal is to convert 71 2/3 into an improper fraction.

• Lowest Terms (Simplified Fraction): A fraction is in its lowest terms, or simplified, when the greatest common divisor (GCD) of the numerator and denominator is 1. This means there's no whole number other than 1 that can divide both the numerator and denominator evenly.

Converting 71 2/3 to an Improper Fraction

The conversion process involves two main steps:

Step 1: Multiply the whole number by the denominator.

In our example, the whole number is 71, and the denominator of the fraction is 3. Therefore, we multiply 71 by 3:

71 * 3 = 213

Step 2: Add the numerator to the result from Step 1.

The numerator of our fraction is 2. We add this to the result from Step 1 (213):

213 + 2 = 215

Step 3: Keep the same denominator.

The denominator remains unchanged throughout the conversion. Therefore, our denominator stays as 3.

Step 4: Combine the results to form the improper fraction.

Combining the results from Step 2 (215) and Step 3 (3), we get the improper fraction:

215/3

This is the improper fraction representation of the mixed number 71 2/3.

Simplifying the Improper Fraction (Finding the Lowest Terms)

While 215/3 is an improper fraction representing the same value as 71 2/3, we need to check if it's in its lowest terms. To do this, we find the greatest common divisor (GCD) of the numerator (215) and the denominator (3).

The GCD is the largest number that divides both 215 and 3 without leaving a remainder. Since 3 is a prime number (only divisible by 1 and itself), and 3 does not divide 215 evenly (215 divided by 3 leaves a remainder of 1), the GCD of 215 and 3 is 1.

Because the GCD is 1, the fraction 215/3 is already in its lowest terms. Therefore, the final answer is:

215/3

Practical Applications and Further Examples

Converting mixed numbers to improper fractions is essential in various mathematical operations, including:

• Addition and Subtraction of Fractions: It's easier to add or subtract fractions when they have a common denominator. Converting mixed numbers to improper fractions facilitates this process.

• Multiplication and Division of Fractions: Multiplying and dividing fractions often involves simpler calculations when working with improper fractions.

• Solving Equations: Many algebraic equations involve fractions, and converting mixed numbers to improper fractions streamlines the solution process.

• Real-world Applications: Numerous real-world problems, from cooking and construction to engineering and finance, involve fraction calculations, making this skill highly practical.

Let's explore a few more examples:

Example 1: Convert 5 1/4 to an improper fraction in lowest terms.

1. Multiply the whole number by the denominator: 5 * 4 = 20
2. Add the numerator: 20 + 1 = 21
3. Keep the same denominator: 4
4. The improper fraction is 21/4.
5. Find the GCD of 21 and 4: The GCD is 1.
6. The simplified fraction is 21/4.

Example 2: Convert 12 3/5 to an improper fraction in lowest terms.

1. Multiply the whole number by the denominator: 12 * 5 = 60
2. Add the numerator: 60 + 3 = 63
3. Keep the same denominator: 5
4. The improper fraction is 63/5.
5. Find the GCD of 63 and 5: The GCD is 1.
6. The simplified fraction is 63/5.

Example 3: Convert 8 4/8 to an improper fraction in lowest terms.

1. Multiply the whole number by the denominator: 8 * 8 = 64
2. Add the numerator: 64 + 4 = 68
3. Keep the same denominator: 8
4. The improper fraction is 68/8.
5. Find the GCD of 68 and 8: The GCD is 4.
6. Divide both numerator and denominator by the GCD: 68/4 = 17 and 8/4 = 2
7. The simplified fraction is 17/2. Notice here that the original fraction was not in lowest terms.

Conclusion

Converting mixed numbers to improper fractions in lowest terms is a fundamental mathematical process with wide-ranging applications. By following the steps outlined above and practicing with various examples, you can master this skill and confidently tackle more complex fraction problems. Remember to always check for simplification by finding the GCD of the numerator and denominator to express your answer in the most concise and accurate form. This ensures your calculations are efficient and your answers are mathematically sound.