Simplifying Fractions A Step-by-Step Guide With Examples

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In the realm of mathematics, fractions play a fundamental role, representing parts of a whole. However, fractions can often appear in complex forms, making it challenging to grasp their true value. This is where the concept of simplifying fractions comes into play. Simplifying a fraction involves reducing it to its lowest terms, where the numerator and denominator have no common factors other than 1. This process not only makes fractions easier to understand but also simplifies calculations and comparisons.

To effectively simplify fractions, we embark on a journey of mathematical exploration, unraveling the techniques and strategies that empower us to express fractions in their most concise form. This comprehensive guide will delve into the intricacies of fraction simplification, equipping you with the knowledge and skills to confidently tackle any fraction-related challenge.

Understanding the Essence of Fraction Simplification

At the heart of fraction simplification lies the principle of equivalence. Equivalent fractions represent the same proportion or value, even though they may appear different. For instance, the fractions 1/2 and 2/4 are equivalent because they both represent half of a whole. Simplifying a fraction involves finding an equivalent fraction with the smallest possible numerator and denominator.

The key to simplifying fractions is identifying 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. Once the GCF is found, both the numerator and denominator are divided by it, resulting in the simplified fraction.

Let's illustrate this with an example. Consider the fraction 12/18. To simplify this fraction, we first need to find the GCF of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 18 are 1, 2, 3, 6, 9, and 18. The GCF of 12 and 18 is 6. Dividing both the numerator and denominator by 6, we get 12/6 = 2 and 18/6 = 3. Therefore, the simplified form of 12/18 is 2/3.

Mastering the Techniques of Fraction Simplification

Several techniques can be employed to simplify fractions, each offering its own advantages. Let's explore some of the most commonly used methods:

1. Finding the Greatest Common Factor (GCF)

As mentioned earlier, the GCF plays a pivotal role in fraction simplification. To find the GCF, we can list the factors of both the numerator and denominator and identify the largest factor they share. Alternatively, we can use the prime factorization method. This method involves breaking down both the numerator and denominator into their prime factors. The GCF is then found by multiplying the common prime factors raised to the lowest power they appear in either factorization.

For example, let's find the GCF of 30 and 72 using the prime factorization method. The prime factorization of 30 is 2 x 3 x 5, while the prime factorization of 72 is 2 x 2 x 2 x 3 x 3. The common prime factors are 2 and 3. The lowest power of 2 that appears in either factorization is 2^1, and the lowest power of 3 is 3^1. Therefore, the GCF of 30 and 72 is 2 x 3 = 6.

2. Dividing by Common Factors

Once the GCF is determined, we divide both the numerator and denominator by it. This process reduces the fraction to its lowest terms. For instance, if we have the fraction 24/36, we first find the GCF of 24 and 36, which is 12. Dividing both the numerator and denominator by 12, we get 24/12 = 2 and 36/12 = 3. Therefore, the simplified form of 24/36 is 2/3.

3. Repeated Simplification

In some cases, simplifying a fraction may require multiple steps. If the numerator and denominator still have common factors after the initial simplification, we can repeat the process until the fraction is in its lowest terms. For example, let's simplify the fraction 36/48. The GCF of 36 and 48 is 12. Dividing both the numerator and denominator by 12, we get 36/12 = 3 and 48/12 = 4. So, the fraction becomes 3/4. Since 3 and 4 have no common factors other than 1, the fraction is now in its simplest form.

Applying Fraction Simplification: Real-World Scenarios

The ability to simplify fractions extends beyond the classroom, finding practical applications in various real-world scenarios. Consider these examples:

1. Cooking and Baking

Recipes often involve fractional measurements. Simplifying these fractions can make it easier to measure ingredients accurately. For instance, if a recipe calls for 4/8 cup of flour, simplifying this fraction to 1/2 cup makes the measurement more straightforward.

2. Sharing and Dividing

When sharing objects or dividing quantities, fractions often arise. Simplifying these fractions ensures fair and accurate distribution. For example, if you want to divide a pizza into 12 slices and share it among 4 people, each person would receive 3/12 of the pizza. Simplifying this fraction to 1/4 makes it clear that each person gets one-fourth of the pizza.

3. Understanding Proportions

Fractions are essential for understanding proportions and ratios. Simplifying fractions helps to compare different proportions easily. For example, if one store offers a discount of 20/100 on an item, while another store offers a discount of 1/5, simplifying both fractions to 1/5 reveals that the discounts are equivalent.

Practice Problems: Sharpening Your Fraction Simplification Skills

To solidify your understanding of fraction simplification, let's tackle a series of practice problems:

  1. Simplify the fraction 15/25.
  2. Simplify the fraction 28/42.
  3. Simplify the fraction 36/60.
  4. Simplify the fraction 45/75.
  5. Simplify the fraction 60/90.

Solutions:

  1. 3/5
  2. 2/3
  3. 3/5
  4. 3/5
  5. 2/3

Conclusion: Mastering the Art of Fraction Simplification

Simplifying fractions is a fundamental skill in mathematics, empowering us to express fractions in their most concise and understandable form. By mastering the techniques and strategies discussed in this guide, you can confidently tackle any fraction-related challenge, whether it's in the classroom or in real-world scenarios.

Remember, practice makes perfect. The more you practice simplifying fractions, the more proficient you will become. So, embrace the challenge, explore the world of fractions, and unlock the power of simplification.

Now, let's apply our knowledge to the specific fractions provided and simplify each one to its lowest terms:

  1. 7/14
  • Identify the Greatest Common Factor (GCF): The GCF of 7 and 14 is 7.
  • Divide both numerator and denominator by the GCF: 7 ÷ 7 = 1 and 14 ÷ 7 = 2
  • Simplified Fraction: 1/2

Simplifying fractions is a crucial skill in mathematics, allowing us to express ratios and proportions in their most concise form. When we encounter a fraction like 7/14, the goal is to find an equivalent fraction where the numerator and denominator share no common factors other than 1. This process not only makes the fraction easier to understand but also simplifies calculations and comparisons. In this case, we begin by identifying the greatest common factor (GCF) of 7 and 14. The factors of 7 are 1 and 7, while the factors of 14 are 1, 2, 7, and 14. The largest factor they share is 7. Next, we divide both the numerator and the denominator by this GCF. Dividing 7 by 7 gives us 1, and dividing 14 by 7 gives us 2. Thus, the simplified fraction is 1/2. This means that 7/14 and 1/2 represent the same proportion, but 1/2 is in its simplest form, making it easier to visualize and work with. Mastering the simplification of fractions like this one lays a strong foundation for more advanced mathematical concepts. The ability to quickly and accurately simplify fractions is invaluable in various fields, from everyday calculations to complex engineering problems. By reducing a fraction to its lowest terms, we not only make it easier to understand but also minimize the chances of errors in further calculations. This systematic approach ensures clarity and precision in mathematical operations. The concept of simplifying fractions is deeply rooted in the principle of equivalence, where different fractions can represent the same value. Understanding this principle helps in grasping the fundamental nature of fractions and their applications. As we progress in mathematics, the ability to simplify fractions becomes an essential tool for solving more complex problems involving algebraic expressions, ratios, and proportions. The process of simplifying fractions also reinforces the understanding of factors and multiples, which are key building blocks in number theory. In essence, mastering fraction simplification is not just about reducing numbers; it's about developing a deeper understanding of mathematical relationships and their applications in the real world.

  1. 4/20
  • Identify the Greatest Common Factor (GCF): The GCF of 4 and 20 is 4.
  • Divide both numerator and denominator by the GCF: 4 ÷ 4 = 1 and 20 ÷ 4 = 5
  • Simplified Fraction: 1/5

When simplifying the fraction 4/20, we are essentially looking for the simplest way to express the ratio between 4 and 20. The process begins by identifying the greatest common factor (GCF) of these two numbers. The factors of 4 are 1, 2, and 4, while the factors of 20 are 1, 2, 4, 5, 10, and 20. The largest number that appears in both lists is 4, making it the GCF. Next, we divide both the numerator and the denominator by the GCF. Dividing 4 by 4 results in 1, and dividing 20 by 4 results in 5. Thus, the simplified fraction is 1/5. This means that 4/20 and 1/5 are equivalent fractions, representing the same proportion. However, 1/5 is in its simplest form, which makes it easier to visualize and use in calculations. For instance, if we imagine a pie cut into 20 slices, 4/20 represents 4 of those slices. In contrast, 1/5 represents one-fifth of the pie. Both represent the same amount, but 1/5 gives a clearer sense of the proportion. The skill of simplifying fractions is fundamental in various mathematical contexts, including algebra, geometry, and calculus. It helps in reducing complex expressions to manageable forms, making problem-solving more efficient. The process also reinforces the understanding of divisibility and factors, which are essential concepts in number theory. In practical terms, simplifying fractions is useful in everyday situations. For example, if you have 40 apples and want to divide them equally among 20 people, each person would get 4/20 of the total apples, which simplifies to 1/5. Understanding this concept helps in making quick decisions and ensuring fair distribution. The ability to simplify fractions also contributes to mathematical fluency, which is the ability to solve problems accurately and efficiently. A fluent mathematician can quickly identify the GCF and simplify fractions mentally, saving time and reducing the likelihood of errors. In summary, simplifying fractions like 4/20 to 1/5 is more than just a mechanical process; it’s a way to develop a deeper understanding of numerical relationships and proportions, which is crucial for success in mathematics and real-world applications.

  1. 14/21
  • Identify the Greatest Common Factor (GCF): The GCF of 14 and 21 is 7.
  • Divide both numerator and denominator by the GCF: 14 ÷ 7 = 2 and 21 ÷ 7 = 3
  • Simplified Fraction: 2/3

Simplifying the fraction 14/21 is a classic example of how reducing fractions to their lowest terms can clarify their value. The first step in this process is to identify the greatest common factor (GCF) of the numerator and the denominator. The factors of 14 are 1, 2, 7, and 14, while the factors of 21 are 1, 3, 7, and 21. The largest number that both 14 and 21 share as a factor is 7. Once we have identified the GCF, we divide both the numerator and the denominator by it. Dividing 14 by 7 gives us 2, and dividing 21 by 7 gives us 3. Therefore, the simplified fraction is 2/3. This means that the fractions 14/21 and 2/3 are equivalent, representing the same proportion or value. However, 2/3 is in its simplest form, making it easier to understand and use in calculations. The process of simplifying fractions helps in various mathematical contexts. For example, when comparing fractions, it is often easier to compare them when they are in their simplest forms. Similarly, when performing operations such as addition or subtraction with fractions, simplifying the fractions first can make the calculations more manageable. Understanding how to simplify fractions also builds a strong foundation for more advanced mathematical concepts, such as algebraic expressions and equations. In real-world applications, simplifying fractions can be useful in situations such as cooking, where recipes often involve fractional measurements. By simplifying these fractions, we can more accurately measure ingredients and achieve the desired results. For instance, if a recipe calls for 14/21 of a cup of sugar, simplifying this to 2/3 of a cup makes the measurement more straightforward. Furthermore, the skill of simplifying fractions helps in developing a deeper understanding of numerical relationships and proportions. It reinforces the concept that different fractions can represent the same value, and it encourages us to look for the most efficient way to express these values. In conclusion, simplifying 14/21 to 2/3 is a fundamental mathematical skill that has practical applications in various aspects of life. It not only simplifies calculations but also enhances our understanding of fractions and their relationships.

  1. 12/21
  • Identify the Greatest Common Factor (GCF): The GCF of 12 and 21 is 3.
  • Divide both numerator and denominator by the GCF: 12 ÷ 3 = 4 and 21 ÷ 3 = 7
  • Simplified Fraction: 4/7

To simplify the fraction 12/21, we again turn to the concept of finding the greatest common factor (GCF). The GCF is the largest number that divides both the numerator and the denominator without leaving a remainder. In this case, the numerator is 12, and the denominator is 21. Let's list the factors of each number. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 21 are 1, 3, 7, and 21. By comparing these lists, we can see that the largest number that appears in both is 3. Therefore, the GCF of 12 and 21 is 3. Now that we have identified the GCF, we divide both the numerator and the denominator by this number. Dividing 12 by 3 gives us 4, and dividing 21 by 3 gives us 7. Thus, the simplified fraction is 4/7. This means that 12/21 and 4/7 are equivalent fractions. They represent the same proportion, but 4/7 is in its simplest form because 4 and 7 have no common factors other than 1. Simplifying fractions is an important skill in mathematics for several reasons. First, it makes fractions easier to understand and visualize. For example, it's easier to picture 4/7 of a pie than 12/21 of a pie. Second, simplified fractions are easier to work with in calculations. When adding, subtracting, multiplying, or dividing fractions, it's often helpful to simplify them first to reduce the size of the numbers involved. Third, simplifying fractions helps in comparing fractions. If we have two fractions that we want to compare, it's easier to do so if they are both in their simplest forms. In real-world applications, simplifying fractions can be useful in various situations. For example, if we are measuring ingredients for a recipe, we might need to simplify a fraction to determine the correct amount. Or, if we are dividing something into equal parts, we might need to simplify a fraction to ensure that each part is the same size. In summary, simplifying 12/21 to 4/7 is a fundamental mathematical skill that has both theoretical and practical applications. It helps us to understand fractions better, to perform calculations more easily, and to solve real-world problems more effectively.

  1. 12/18
  • Identify the Greatest Common Factor (GCF): The GCF of 12 and 18 is 6.
  • Divide both numerator and denominator by the GCF: 12 ÷ 6 = 2 and 18 ÷ 6 = 3
  • Simplified Fraction: 2/3

Simplifying the fraction 12/18 involves finding an equivalent fraction with the smallest possible numerator and denominator. To achieve this, we need to identify the greatest common factor (GCF) of 12 and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 18 are 1, 2, 3, 6, 9, and 18. Comparing these lists, we find that the largest number that appears in both is 6, making it the GCF. Once we have determined the GCF, we divide both the numerator and the denominator by it. Dividing 12 by 6 gives us 2, and dividing 18 by 6 gives us 3. Consequently, the simplified fraction is 2/3. This means that 12/18 and 2/3 represent the same proportion or value, but 2/3 is in its simplest form, which is often easier to work with. Simplifying fractions is a fundamental skill in mathematics with wide-ranging applications. It helps us to express quantities in their most understandable form, making it easier to grasp their magnitude. In practical situations, simplifying fractions can be invaluable. For instance, if we are sharing a pizza with 18 slices among 12 people, the fraction 12/18 represents the proportion of the pizza that is being shared. By simplifying this fraction to 2/3, we can easily see that each person is receiving two-thirds of a slice. Simplifying fractions also plays a crucial role in more advanced mathematical concepts. In algebra, for example, simplifying fractions is often necessary when working with algebraic expressions and equations. It helps to reduce the complexity of the expressions, making them easier to manipulate and solve. In geometry, simplifying fractions can be useful when calculating ratios and proportions. The ability to simplify fractions accurately and efficiently is a hallmark of mathematical fluency. It demonstrates a strong understanding of factors, multiples, and divisibility, which are essential building blocks for more advanced topics. The process of simplifying fractions also reinforces the concept of equivalence, where different fractions can represent the same value. This understanding is crucial for developing a deep and flexible understanding of fractions. In conclusion, simplifying 12/18 to 2/3 is a fundamental mathematical skill that has both practical and theoretical significance. It helps us to understand quantities more clearly, to solve problems more efficiently, and to build a strong foundation for more advanced mathematical concepts.

  1. 5/50
  • Identify the Greatest Common Factor (GCF): The GCF of 5 and 50 is 5.
  • Divide both numerator and denominator by the GCF: 5 ÷ 5 = 1 and 50 ÷ 5 = 10
  • Simplified Fraction: 1/10

Simplifying the fraction 5/50 involves reducing it to its lowest terms, making it easier to understand and work with. The first step in this process is to identify the greatest common factor (GCF) of the numerator and the denominator. In this case, the numerator is 5, and the denominator is 50. To find the GCF, we list the factors of each number. The factors of 5 are 1 and 5. The factors of 50 are 1, 2, 5, 10, 25, and 50. The largest number that appears in both lists is 5. Therefore, the GCF of 5 and 50 is 5. Once we have identified the GCF, we divide both the numerator and the denominator by it. Dividing 5 by 5 gives us 1, and dividing 50 by 5 gives us 10. Thus, the simplified fraction is 1/10. This means that the fractions 5/50 and 1/10 are equivalent. They represent the same proportion, but 1/10 is in its simplest form, which makes it easier to visualize and compare. For example, if we imagine a group of 50 people and want to represent 5 of them as a fraction of the whole group, we would initially write 5/50. However, by simplifying this fraction to 1/10, we can quickly understand that 5 people represent one-tenth of the group. Simplifying fractions is a crucial skill in mathematics for several reasons. It makes fractions easier to understand, allows for easier comparison of fractions, and simplifies calculations involving fractions. In practical applications, simplifying fractions can be useful in various situations, such as cooking, measuring, and sharing. For instance, if a recipe calls for 5/50 of a cup of an ingredient, we can simplify this to 1/10 of a cup, making it easier to measure accurately. Similarly, if we are dividing a cake into 50 slices and want to give 5 slices to someone, we can simplify the fraction 5/50 to 1/10 to understand that they are receiving one-tenth of the cake. In summary, simplifying 5/50 to 1/10 is a fundamental mathematical operation that enhances our understanding of fractions and their applications in real-world scenarios. It is a skill that is essential for success in mathematics and everyday life.

  1. 30/72
  • Identify the Greatest Common Factor (GCF): The GCF of 30 and 72 is 6.
  • Divide both numerator and denominator by the GCF: 30 ÷ 6 = 5 and 72 ÷ 6 = 12
  • Simplified Fraction: 5/12

Simplifying fractions is a fundamental skill in mathematics, and when we tackle the fraction 30/72, we're essentially seeking the most reduced form that represents the same proportional value. The process begins by identifying the greatest common factor (GCF) of the numerator (30) and the denominator (72). The GCF is the largest number that divides both 30 and 72 without leaving a remainder. To find it, we can list the factors of each number. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. By comparing these lists, we identify the largest number present in both, which is 6. With the GCF determined, we proceed to divide both the numerator and the denominator by 6. This yields 30 ÷ 6 = 5 and 72 ÷ 6 = 12. Consequently, the simplified fraction is 5/12. This means that 30/72 and 5/12 are equivalent fractions, representing the same proportional quantity. However, 5/12 is in its simplest form, as 5 and 12 share no common factors other than 1. Simplifying fractions is not merely an academic exercise; it has practical applications in numerous real-world scenarios. For instance, consider a scenario where you have 30 identical items and wish to divide them into 72 equal parts. The fraction 30/72 represents the size of each part relative to the whole. By simplifying it to 5/12, you gain a clearer understanding of the proportion involved. This skill is particularly useful in cooking, where recipes often involve fractional measurements. It also plays a crucial role in various mathematical contexts. When adding or subtracting fractions, simplifying them first can make the calculations much easier. In algebra, simplifying fractions is essential for working with rational expressions and solving equations. The ability to quickly and accurately simplify fractions is a valuable asset in any mathematical endeavor. It demonstrates a strong grasp of fundamental concepts and enhances one's problem-solving abilities. In conclusion, simplifying 30/72 to 5/12 exemplifies the power of mathematical reduction, providing a clearer representation of the underlying proportion and paving the way for easier calculations and better comprehension.

  1. 4/40
  • Identify the Greatest Common Factor (GCF): The GCF of 4 and 40 is 4.
  • Divide both numerator and denominator by the GCF: 4 ÷ 4 = 1 and 40 ÷ 4 = 10
  • Simplified Fraction: 1/10

When we encounter the fraction 4/40, simplifying it involves expressing it in its most reduced form, which means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. To do this, we need to identify the greatest common factor (GCF) of 4 and 40. The GCF is the largest number that divides both 4 and 40 without leaving a remainder. Let's list the factors of each number. The factors of 4 are 1, 2, and 4. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. By comparing these lists, we can see that the largest number that appears in both is 4. Therefore, the GCF of 4 and 40 is 4. Now that we have identified the GCF, we divide both the numerator and the denominator by it. Dividing 4 by 4 gives us 1, and dividing 40 by 4 gives us 10. Thus, the simplified fraction is 1/10. This means that the fractions 4/40 and 1/10 are equivalent. They represent the same proportion, but 1/10 is in its simplest form. Simplifying fractions is a fundamental skill in mathematics that has numerous applications. It makes fractions easier to understand, compare, and work with in calculations. For example, if we are sharing something equally among 40 people and each person receives 4 parts, the fraction 4/40 represents the proportion each person receives. By simplifying this fraction to 1/10, we can quickly grasp that each person receives one-tenth of the whole. In practical situations, simplifying fractions can be particularly useful. For instance, in cooking, recipes often involve fractional measurements. Simplifying these fractions can make it easier to measure ingredients accurately. Similarly, in construction or engineering, simplifying fractions can help in calculating dimensions and proportions. The ability to simplify fractions also lays a strong foundation for more advanced mathematical concepts. When working with algebraic expressions and equations, simplifying fractions is often a necessary step in solving problems. In summary, simplifying 4/40 to 1/10 is a fundamental mathematical skill that enhances our understanding of fractions and their applications in various contexts. It simplifies calculations, facilitates comparisons, and builds a solid foundation for more advanced mathematical concepts.

  1. 12/30
  • Identify the Greatest Common Factor (GCF): The GCF of 12 and 30 is 6.
  • Divide both numerator and denominator by the GCF: 12 ÷ 6 = 2 and 30 ÷ 6 = 5
  • Simplified Fraction: 2/5

To simplify the fraction 12/30, we embark on the process of finding an equivalent fraction that is expressed in its lowest terms. This involves identifying the greatest common factor (GCF) of the numerator and the denominator. In this case, the numerator is 12, and the denominator is 30. The GCF is the largest number that divides both 12 and 30 without leaving a remainder. Let's list the factors of each number to determine the GCF. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. By comparing these lists, we can see that the largest number that appears in both is 6. Therefore, the GCF of 12 and 30 is 6. Once we have identified the GCF, we divide both the numerator and the denominator by it. Dividing 12 by 6 gives us 2, and dividing 30 by 6 gives us 5. Thus, the simplified fraction is 2/5. This means that 12/30 and 2/5 are equivalent fractions. They represent the same proportion, but 2/5 is in its simplest form because 2 and 5 have no common factors other than 1. Simplifying fractions is a crucial skill in mathematics, with wide-ranging applications. It makes fractions easier to understand and visualize, which is particularly helpful when comparing fractions or performing calculations. For example, it is easier to conceptualize 2/5 of a pie than 12/30 of a pie. In practical situations, simplifying fractions can be invaluable. For instance, if a recipe calls for 12/30 of a cup of flour, simplifying this to 2/5 of a cup makes it easier to measure the correct amount. Similarly, when working with ratios and proportions, simplifying fractions can help to clarify the relationships between quantities. The ability to simplify fractions is also essential for more advanced mathematical concepts. In algebra, simplifying fractions is often a necessary step when solving equations or working with rational expressions. In summary, simplifying 12/30 to 2/5 is a fundamental mathematical operation that enhances our understanding of fractions and their applications in various contexts. It is a skill that is essential for success in mathematics and everyday life.

  1. 30/55
  • Identify the Greatest Common Factor (GCF): The GCF of 30 and 55 is 5.
  • Divide both numerator and denominator by the GCF: 30 ÷ 5 = 6 and 55 ÷ 5 = 11
  • Simplified Fraction: 6/11

Simplifying the fraction 30/55 is a process that involves reducing it to its lowest terms while maintaining its value. The key to this process is finding the greatest common factor (GCF) of the numerator and the denominator. In this case, the numerator is 30, and the denominator is 55. To identify the GCF, we need to list the factors of both numbers. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 55 are 1, 5, 11, and 55. Comparing these two lists, we can see that the largest number that appears in both is 5. Therefore, the GCF of 30 and 55 is 5. Once we have determined the GCF, we divide both the numerator and the denominator by this number. Dividing 30 by 5 gives us 6, and dividing 55 by 5 gives us 11. As a result, the simplified fraction is 6/11. This means that the fractions 30/55 and 6/11 are equivalent; they represent the same proportion or value. However, 6/11 is in its simplest form, which makes it easier to understand and use in various mathematical contexts. Simplifying fractions is a crucial skill in mathematics. It allows us to express quantities in their most concise form, making them easier to compare and work with. For instance, if we were comparing the fractions 30/55 and another fraction, it would be more straightforward to compare 6/11 with the other fraction. In practical applications, simplifying fractions is invaluable. For example, if we are sharing a quantity, such as a pizza, among a group of people, simplifying the fraction that represents each person's share can make it easier to understand the distribution. If the pizza is cut into 55 slices and each person receives 30 slices, the fraction 30/55 represents each person's share. Simplifying this to 6/11 provides a clearer understanding of the proportion. Furthermore, the ability to simplify fractions is essential for more advanced mathematical concepts. When solving algebraic equations, simplifying fractions can reduce the complexity of the equation and make it easier to solve. In calculus, simplifying fractions is often a necessary step in evaluating limits and derivatives. In summary, simplifying 30/55 to 6/11 is a fundamental mathematical skill that has both practical and theoretical significance. It enhances our understanding of fractions, simplifies calculations, and lays a strong foundation for more advanced mathematical topics.