What Is 7 8 7/16 Reduced To The Lowest Terms

What is 7 8 7/16 Reduced to the Lowest Terms? A Comprehensive Guide to Fraction Simplification

Understanding how to reduce fractions to their lowest terms is a fundamental skill in mathematics. This seemingly simple process is crucial for various applications, from basic arithmetic to advanced calculus. This comprehensive guide will delve into the step-by-step process of simplifying the mixed number 7 8 7/16, explaining the underlying principles and providing practical examples to solidify your understanding. We'll also explore related concepts and techniques to enhance your overall fraction manipulation skills.

Understanding Mixed Numbers and Improper Fractions

Before we tackle the simplification of 7 8 7/16, let's clarify the terminology. A mixed number combines a whole number and a proper fraction (a fraction where the numerator is smaller than the denominator). In our case, 7 8 7/16 is a mixed number where 7 is the whole number, and 8 7/16 is the fractional part.

An improper fraction, on the other hand, has a numerator that is greater than or equal to its denominator. To simplify a mixed number, we first convert it into an improper fraction. This is achieved by multiplying the whole number by the denominator and then adding the numerator. The result becomes the new numerator, while the denominator remains the same.

Let's convert 7 8 7/16 into an improper fraction:

1. Multiply the whole number by the denominator: 7 * 16 = 112
2. Add the numerator: 112 + 7 = 119
3. Keep the denominator: 16

Therefore, 7 8 7/16 is equivalent to the improper fraction 119/16.

Reducing Fractions to Lowest Terms: The Greatest Common Divisor (GCD)

Reducing a fraction to its lowest terms means simplifying it to its smallest equivalent form. This is done by finding the greatest common divisor (GCD), also known as the highest common factor (HCF), of the numerator and the denominator. The GCD is the largest number that divides both the numerator and denominator without leaving a remainder.

Once the GCD is found, both the numerator and denominator are divided by the GCD. This results in a simplified fraction that represents the same value as the original fraction but in its most compact form.

Several methods can be used to find the GCD:

1. Listing Factors:

This method involves listing all the factors of the numerator and denominator and identifying the largest common factor.

Factors of 119: 1, 7, 17, 119 Factors of 16: 1, 2, 4, 8, 16

The largest common factor is 1.

2. Prime Factorization:

This is a more systematic approach. We find the prime factorization of both the numerator and denominator. The GCD is then the product of the common prime factors raised to the lowest power.

Prime factorization of 119: 7 x 17 Prime factorization of 16: 2 x 2 x 2 x 2 = 2⁴

There are no common prime factors between 119 and 16.

3. Euclidean Algorithm:

This is an efficient algorithm, especially for larger 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 to 119 and 16:

119 = 16 * 7 + 7 16 = 7 * 2 + 2 7 = 2 * 3 + 1 2 = 1 * 2 + 0

The last non-zero remainder is 1. Therefore, the GCD of 119 and 16 is 1.

Simplifying 119/16: The Final Result

Since the GCD of 119 and 16 is 1, the fraction 119/16 is already in its lowest terms. It cannot be simplified further. Therefore, the simplified form of 7 8 7/16 is 7 7/16. While we converted it to an improper fraction for the simplification process, the final answer is best expressed as a mixed number for clarity and ease of understanding.

Further Exploration: Working with Fractions

Mastering fraction simplification is a building block for more advanced mathematical concepts. Here are some related areas to explore:

• Adding and Subtracting Fractions: To perform these operations, you must have a common denominator. Simplifying fractions after addition or subtraction often leads to a more concise result.
• Multiplying and Dividing Fractions: These operations involve multiplying numerators and denominators directly (multiplication) or inverting the second fraction and multiplying (division). Simplification plays a crucial role in obtaining the simplest form of the answer.
• Converting Decimals to Fractions: Understanding fraction simplification helps in converting decimals into their simplest fractional form.
• Solving Equations Involving Fractions: In algebraic equations, simplifying fractions makes the equation easier to solve.

Conclusion: The Importance of Fraction Simplification

Simplifying fractions is a fundamental skill essential for various mathematical operations and applications. The process involves finding the greatest common divisor of the numerator and denominator and dividing both by this GCD. This guide has comprehensively covered the simplification of 7 8 7/16, demonstrating the step-by-step process and explaining the underlying concepts. Mastering fraction simplification not only improves your mathematical abilities but also lays a solid foundation for tackling more complex mathematical problems. Remember to practice regularly to enhance your proficiency and confidence in working with fractions. The more you practice, the faster and more intuitive the process will become.