Simplify The Following Expression And Provide The Answer As A Proper Fraction Or Mixed Number: $\frac{4\frac{5}{7}}{3\frac{3}{4}} = $

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In the realm of mathematics, fractions and mixed numbers form the bedrock of many numerical operations. Mastering the art of simplifying these expressions is not only crucial for academic success but also for practical applications in everyday life. In this comprehensive guide, we will delve into the intricacies of simplifying fractions and mixed numbers, equipping you with the knowledge and skills to tackle these challenges with confidence. We will dissect the process of converting mixed numbers to improper fractions, simplifying improper fractions to their lowest terms, and performing various arithmetic operations on fractions. So, buckle up and embark on this enlightening journey into the world of fractions and mixed numbers!

Understanding Fractions and Mixed Numbers

Before we embark on the simplification journey, let's lay a solid foundation by understanding the basic concepts of fractions and mixed numbers. A fraction represents a part of a whole, denoted by two numbers separated by a line. The number above the line is the numerator, representing the number of parts we have, and the number below the line is the denominator, indicating the total number of equal parts the whole is divided into. For instance, the fraction 3/4 signifies that we have 3 parts out of a whole divided into 4 equal parts.

Now, let's turn our attention to mixed numbers. A mixed number combines a whole number and a fraction. It represents a quantity that is greater than a whole but less than the next whole number. For example, the mixed number 2 1/2 represents two whole units and an additional one-half unit. Understanding the composition of mixed numbers is crucial for converting them into improper fractions, a key step in simplifying expressions involving fractions.

Converting Mixed Numbers to Improper Fractions

The conversion of mixed numbers to improper fractions is a fundamental step in simplifying expressions involving mixed numbers. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we follow a systematic approach. First, we multiply the whole number part of the mixed number by the denominator of the fractional part. Then, we add the numerator of the fractional part to the product obtained in the previous step. This sum becomes the numerator of the improper fraction. The denominator of the improper fraction remains the same as the denominator of the fractional part of the mixed number.

Let's illustrate this process with an example. Consider the mixed number 3 2/5. To convert this to an improper fraction, we multiply the whole number 3 by the denominator 5, which gives us 15. Then, we add the numerator 2 to this product, resulting in 17. Therefore, the improper fraction equivalent of 3 2/5 is 17/5. Mastering this conversion technique is essential for performing arithmetic operations involving mixed numbers and simplifying complex fractional expressions.

Simplifying Improper Fractions

Once we have converted mixed numbers to improper fractions, the next step is to simplify these improper fractions to their lowest terms. An improper fraction is considered simplified when the numerator and denominator have no common factors other than 1. To simplify an improper fraction, we need to find the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest number that divides both the numerator and denominator without leaving a remainder. There are various methods to find the GCF, such as listing factors, prime factorization, and the Euclidean algorithm.

Once we have determined the GCF, we divide both the numerator and denominator by the GCF. This process reduces the fraction to its simplest form. For instance, let's consider the improper fraction 24/18. To simplify this, we first find the GCF of 24 and 18, which is 6. Then, we divide both the numerator and denominator by 6, resulting in the simplified fraction 4/3. Simplifying improper fractions is crucial for expressing fractions in their most concise and manageable form, facilitating further calculations and comparisons.

Dividing Fractions: A Step-by-Step Guide

Dividing fractions might seem daunting at first, but with the right approach, it becomes a straightforward process. The key to dividing fractions lies in the concept of reciprocals. The reciprocal of a fraction is obtained by swapping the numerator and denominator. For instance, the reciprocal of 2/3 is 3/2.

To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction. This effectively transforms the division problem into a multiplication problem, which we can solve using the standard multiplication rules for fractions. Let's illustrate this with an example. Suppose we want to divide 3/4 by 1/2. The reciprocal of 1/2 is 2/1. Therefore, we multiply 3/4 by 2/1, which gives us 6/4. This fraction can be further simplified to 3/2. Mastering the division of fractions is essential for various mathematical applications, including solving proportions, scaling recipes, and calculating rates.

Solving the Fraction Problem: A Step-by-Step Solution

Now, let's tackle the original problem presented: Simplify and express the answer as a proper fraction or mixed number (enter only the last value obtained in the answer). $ rac{4 rac{5}{7}}{3 rac{3}{4}} = $

To solve this problem, we will follow a step-by-step approach, applying the techniques we have discussed so far:

  1. Convert mixed numbers to improper fractions:
    • 4 rac{5}{7} = rac{(4 imes 7) + 5}{7} = rac{28 + 5}{7} = rac{33}{7}
    • 3 rac{3}{4} = rac{(3 imes 4) + 3}{4} = rac{12 + 3}{4} = rac{15}{4}
  2. Rewrite the division problem:

The original problem can be rewritten as:

rac{33/7}{15/4}

This represents dividing the fraction 33/7 by the fraction 15/4.

  1. Divide fractions by multiplying by the reciprocal:

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of 15/4 is 4/15. Therefore:

rac{33}{7} imes rac{4}{15} = rac{33 imes 4}{7 imes 15} = rac{132}{105}

  1. Simplify the resulting fraction:

Now, we need to simplify the improper fraction 132/105. To do this, we find the greatest common factor (GCF) of 132 and 105. The GCF of 132 and 105 is 3. Divide both the numerator and denominator by 3:

rac{132 div 3}{105 div 3} = rac{44}{35}

The fraction 44/35 is still an improper fraction, as the numerator is greater than the denominator.

  1. Convert the improper fraction to a mixed number:

To convert the improper fraction 44/35 to a mixed number, we divide the numerator (44) by the denominator (35). The quotient is the whole number part, and the remainder is the numerator of the fractional part. The denominator remains the same.

44 divided by 35 is 1 with a remainder of 9. Therefore, the mixed number is:

1 rac{9}{35}

Therefore, the simplified answer, expressed as a mixed number, is 1 9/35.

Conclusion

In this comprehensive guide, we have explored the intricacies of simplifying fractions and mixed numbers. We have delved into the conversion of mixed numbers to improper fractions, simplified improper fractions to their lowest terms, and mastered the art of dividing fractions. By following the step-by-step approach outlined in this guide, you can confidently tackle a wide range of fraction-related problems. Remember, practice makes perfect, so keep honing your skills and you'll become a fraction simplification maestro in no time! With a solid understanding of fractions and mixed numbers, you'll be well-equipped to excel in mathematics and tackle real-world problems that involve these fundamental concepts.