Expressing (13/15 - 7/10) As A Percentage A Step-by-Step Guide

In the realm of mathematics, converting fractions to percentages is a fundamental skill with practical applications across various fields. This article delves into the process of expressing the result of the subtraction $(\frac{13}{15} - \frac{7}{10})$ as a percentage. We will break down each step, ensuring clarity and a thorough understanding of the underlying concepts. This comprehensive guide is designed to help students, educators, and anyone interested in enhancing their mathematical proficiency. Understanding how to convert fractions to percentages is crucial not only for academic success but also for real-world applications, such as calculating discounts, understanding financial data, and interpreting statistical information. Mastering this skill can significantly improve your quantitative reasoning and problem-solving abilities.

Understanding Fractions and Percentages

Before we dive into the specifics of our problem, let's establish a solid foundation by revisiting the concepts of fractions and percentages. Fractions represent parts of a whole, expressed as a ratio of two numbers: the numerator (the top number) and the denominator (the bottom number). The numerator indicates the number of parts we have, while the denominator indicates the total number of parts the whole is divided into. For example, in the fraction $\frac{13}{15}$, 13 is the numerator, and 15 is the denominator. This means we have 13 parts out of a total of 15. Fractions can represent various quantities, such as proportions, ratios, or probabilities. They are a fundamental building block in mathematics and are used extensively in algebra, calculus, and statistics. Understanding fractions is essential for performing arithmetic operations like addition, subtraction, multiplication, and division, as well as for comparing quantities and solving equations. Percentages, on the other hand, are a way of expressing a number as a fraction of 100. The term "percent" comes from the Latin "per centum," meaning "out of one hundred." A percentage is denoted by the symbol %, so 50% means 50 out of 100. Percentages are widely used to express proportions, rates, and changes. They are particularly useful for comparing quantities of different sizes or scales. For instance, expressing a sales discount as a percentage (e.g., 20% off) makes it easy to understand the proportion of the original price that is being reduced. Similarly, percentages are used in finance to represent interest rates, investment returns, and inflation rates. Converting between fractions and percentages is a crucial skill in mathematics. To convert a fraction to a percentage, we first divide the numerator by the denominator to obtain a decimal. Then, we multiply the decimal by 100 and add the percent symbol (%). Conversely, to convert a percentage to a fraction, we divide the percentage by 100 and simplify the resulting fraction. This understanding of fractions and percentages forms the basis for tackling more complex mathematical problems and real-world applications.

Step-by-Step Solution

Now, let's tackle the problem at hand: expressing $(\frac{13}{15} - \frac{7}{10})$ as a percentage. This involves a series of steps, each building upon the previous one. Step 1: Find a Common Denominator Before we can subtract the fractions, we need to find a common denominator. This is the smallest number that both denominators (15 and 10) can divide into evenly. The least common multiple (LCM) of 15 and 10 is 30. To find the LCM, we can list the multiples of each number until we find a common one: Multiples of 15: 15, 30, 45, 60, ... Multiples of 10: 10, 20, 30, 40, ... The smallest number that appears in both lists is 30, so the LCM of 15 and 10 is 30. Using a common denominator is essential for performing arithmetic operations on fractions because it ensures that we are adding or subtracting like parts. When fractions have different denominators, they represent parts of different sizes, making it impossible to directly add or subtract them. By finding a common denominator, we can rewrite the fractions so that they represent parts of the same size, allowing us to perform the necessary operations. Step 2: Convert the Fractions Next, we need to convert both fractions to have the common denominator of 30. To convert $\frac{13}{15}$ to a fraction with a denominator of 30, we multiply both the numerator and the denominator by 2: $ \frac{13}{15} \times \frac{2}{2} = \frac{26}{30}

Similarly, to convert $\frac{7}{10}$ to a fraction with a denominator of 30, we multiply both the numerator and the denominator by 3:

\frac{7}{10} \times \frac{3}{3} = \frac{21}{30}

$$Converting fractions to equivalent forms with a common denominator is a critical step in many mathematical problems. It allows us to compare fractions, add and subtract them, and solve equations involving fractions. This process involves multiplying both the numerator and the denominator of a fraction by the same number, which does not change the value of the fraction but alters its representation. **Step 3: Subtract the Fractions** Now that both fractions have the same denominator, we can subtract them:$$

\frac{26}{30} - \frac{21}{30} = \frac{26 - 21}{30} = \frac{5}{30}

Subtracting fractions with a common denominator is straightforward: we simply subtract the numerators and keep the denominator the same. This operation gives us the difference between the two fractions, which represents the amount by which one fraction exceeds the other. **Step 4: Simplify the Fraction** The fraction $\frac{5}{30}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 5:

\frac{5}{30} = \frac{5 \div 5}{30 \div 5} = \frac{1}{6}

Simplifying fractions is an important practice in mathematics. It involves reducing a fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor. This process makes the fraction easier to understand and work with, as it represents the same value using the smallest possible numbers. **Step 5: Convert to a Decimal** To convert the fraction $\frac{1}{6}$ to a decimal, we divide the numerator (1) by the denominator (6):

1 \div 6 = 0.1666...

$$The result is a repeating decimal, approximately 0.1667 when rounded to four decimal places. Converting fractions to decimals is a fundamental skill in mathematics. It allows us to express fractions in a different form, which is often more convenient for calculations and comparisons. Decimals are also essential in many real-world applications, such as measurements, finance, and scientific calculations. **Step 6: Convert to a Percentage** Finally, to express the decimal as a percentage, we multiply by 100:$$

0.1667 \times 100 = 16.67

So, $(\frac{13}{15} - \frac{7}{10})$ expressed as a percentage is approximately 16.67%. Converting decimals to percentages involves multiplying the decimal by 100 and adding the percent symbol (%). This process effectively scales the decimal value to a proportion out of 100, making it easy to understand and compare relative quantities. Percentages are widely used in various contexts, such as finance, statistics, and everyday life, to represent proportions, rates, and changes. ## Alternative Methods and Insights While the step-by-step method provides a clear path to the solution, exploring alternative methods can deepen our understanding and problem-solving skills. One alternative approach involves using a calculator to directly compute the decimal value of each fraction and then subtracting them. For example, we can calculate $\frac{13}{15}$ as approximately 0.8667 and $\frac{7}{10}$ as 0.7. Subtracting these values gives us 0. 1667, which, when multiplied by 100, yields 16.67%. This method can be quicker for those comfortable with calculator usage, but it's crucial to understand the underlying principles to avoid errors and ensure accuracy. Another insight to consider is the importance of estimation in problem-solving. Before performing the calculations, we can estimate the final percentage. We know that $\frac{13}{15}$ is close to 1 (or 100%), and $\frac{7}{10}$ is 0.7 (or 70%). Therefore, the difference should be around 30% of $\frac{1}{2}$ (since 7/10 is relatively big, more than a half, so the result will be less than 30%). After subtracting 70% from a value close to 90%, so the result should be significantly less than 30%, meaning the final percentage should be considerably less. This estimation helps us verify our final answer and identify potential mistakes. Furthermore, understanding the relationship between fractions, decimals, and percentages is crucial for mathematical fluency. These different representations of numbers are interconnected, and being able to convert between them effortlessly enhances our ability to solve problems and interpret data. For instance, recognizing that $\frac{1}{6}$ is approximately 16.67% allows us to quickly estimate proportions and make informed decisions in various situations. In addition to these insights, it's worth noting that practice is key to mastering these concepts. The more we work with fractions, decimals, and percentages, the more comfortable and confident we become in handling them. Regular practice not only improves our calculation skills but also deepens our understanding of the underlying principles, allowing us to apply them in diverse contexts. ## Practical Applications The ability to express fractions as percentages is not just a theoretical mathematical skill; it has numerous practical applications in everyday life and various professional fields. In finance, percentages are used to express interest rates, investment returns, and inflation rates. For example, understanding the difference between a 5% annual interest rate and a 5.5% rate can have a significant impact on long-term savings and investments. Similarly, percentages are used to calculate discounts and sales tax, helping consumers make informed purchasing decisions. In statistics, percentages are used to represent proportions and probabilities. For instance, a survey might report that 60% of respondents prefer a particular product, or a weather forecast might state a 30% chance of rain. Understanding these percentages allows us to interpret data and make predictions. In the business world, percentages are used to analyze market share, profit margins, and growth rates. Companies track these metrics to assess their performance and make strategic decisions. For example, a company might aim to increase its market share by 5% or improve its profit margin by 2%. In science and engineering, percentages are used to express concentrations, errors, and efficiencies. For example, a solution might be 10% salt by weight, or an experiment might have a 5% margin of error. Understanding these percentages is crucial for accurate measurements and calculations. In education, percentages are used to calculate grades and assess student performance. A student's score on a test is often expressed as a percentage, providing a clear indication of their understanding of the material. Furthermore, the skill of converting fractions to percentages is valuable in everyday situations such as cooking, where recipes often call for proportions of ingredients, and in home improvement projects, where measurements and ratios are essential. By mastering this skill, we can confidently handle a wide range of practical tasks and make informed decisions in various aspects of our lives. The ability to work with percentages enhances our numerical literacy and empowers us to navigate the quantitative world with greater ease and understanding. ## Conclusion In conclusion, expressing $(\frac{13}{15} - \frac{7}{10})$ as a percentage involves a series of logical steps: finding a common denominator, converting the fractions, subtracting them, simplifying the result, converting to a decimal, and finally, expressing it as a percentage. The result, approximately 16.67%, highlights the importance of understanding fractions, decimals, and percentages, and their interconversion. This skill is not only crucial for academic success but also has wide-ranging applications in real-world scenarios. By mastering these fundamental concepts, we enhance our mathematical proficiency and problem-solving abilities, empowering us to tackle diverse quantitative challenges with confidence. The journey through this problem underscores the value of a step-by-step approach, the importance of estimation, and the interconnectedness of different mathematical concepts. Each step, from finding the common denominator to converting to a percentage, builds upon the previous one, creating a cohesive and logical solution. This process not only provides the answer but also reinforces our understanding of the underlying principles. Furthermore, exploring alternative methods and insights deepens our comprehension and enhances our problem-solving skills. Whether using a calculator or estimating the result, these approaches offer different perspectives and strategies for tackling similar problems in the future. The practical applications of this skill further emphasize its significance. From finance to statistics, from business to everyday life, the ability to convert fractions to percentages is essential for making informed decisions and navigating the quantitative world. By recognizing the relevance of this skill in various contexts, we are motivated to master it and apply it effectively. In essence, the exercise of expressing $(\frac{13}{15} - \frac{7}{10})$ as a percentage is more than just a mathematical problem; it is a journey of learning, understanding, and applying fundamental concepts that empower us to succeed in both academic pursuits and real-world endeavors. As we continue to explore mathematics, we will encounter numerous challenges that require a solid foundation in fractions, decimals, and percentages. By mastering these basics, we pave the way for more advanced topics and unlock the power of quantitative reasoning.