Solving 3/4 - 2/5 A Step-by-Step Guide

Fraction subtraction is a fundamental concept in mathematics, and mastering it is crucial for various real-world applications. This comprehensive guide provides a step-by-step explanation of how to solve the fraction subtraction problem ${\frac{3}{4} - \frac{2}{5}}$. We will break down each step, ensuring a clear understanding of the underlying principles. This exploration will not only help you solve this specific problem but also equip you with the knowledge to tackle similar fraction subtraction challenges with confidence. Let's dive into the intricacies of subtracting fractions and build a solid foundation in this essential mathematical skill. By the end of this guide, you'll be well-versed in the techniques and strategies needed to approach fraction subtraction problems effectively.

Identifying the Challenge: Unlike Denominators

The core challenge in subtracting ${\frac{3}{4}}$ and ${\frac{2}{5}}$ lies in the fact that the fractions have different denominators. The denominator is the bottom number in a fraction, representing the total number of equal parts into which a whole is divided. In our problem, we have denominators of 4 and 5. To subtract fractions, they must have the same denominator. This common denominator represents a unified way of dividing the whole, allowing us to directly compare and subtract the numerators (the top numbers in a fraction, which indicate the number of parts we have). Understanding this requirement is the first crucial step in our journey to solving the problem. Without a common denominator, we cannot accurately determine the difference between the two fractions. Think of it like trying to compare apples and oranges – we need a common unit of measurement to make a meaningful comparison. In the world of fractions, that common unit is the common denominator.

Finding the Least Common Denominator (LCD)

To subtract fractions with unlike denominators, the first crucial step is to find the Least Common Denominator (LCD). The LCD is the smallest multiple that both denominators share. In our case, the denominators are 4 and 5. To find the LCD, we can list the multiples of each denominator:

• Multiples of 4: 4, 8, 12, 16, 20, 24, ...
• Multiples of 5: 5, 10, 15, 20, 25, ...

The smallest multiple that appears in both lists is 20. Therefore, the LCD of 4 and 5 is 20. The LCD serves as the foundation for creating equivalent fractions that we can then subtract. Finding the LCD efficiently is a key skill in fraction operations. There are other methods for finding the LCD, such as prime factorization, but listing multiples is often a straightforward approach, especially for smaller numbers. Mastering the concept of the LCD is essential for successfully subtracting fractions with different denominators.

Creating Equivalent Fractions

Now that we've identified the LCD as 20, our next step is to convert both fractions, ${\frac{3}{4}}$ and ${\frac{2}{5}}$, into equivalent fractions with a denominator of 20. An equivalent fraction represents the same value as the original fraction but has a different numerator and denominator. To create equivalent fractions, we multiply both the numerator and the denominator of the original fraction by the same number. This maintains the fraction's value because we are essentially multiplying by 1.

For ${\frac{3}{4}}$, we need to determine what number we must multiply 4 by to get 20. Since 4 multiplied by 5 equals 20, we multiply both the numerator and the denominator of ${\frac{3}{4}}$ by 5:

${\frac{3}{4} \times \frac{5}{5} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}}$

Similarly, for ${\frac{2}{5}}$, we need to determine what number we must multiply 5 by to get 20. Since 5 multiplied by 4 equals 20, we multiply both the numerator and the denominator of ${\frac{2}{5}}$ by 4:

${\frac{2}{5} \times \frac{4}{4} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}}$

Now, we have successfully converted ${\frac{3}{4}}$ into ${\frac{15}{20}}$ and ${\frac{2}{5}}$ into ${\frac{8}{20}}$. These equivalent fractions share a common denominator, setting the stage for the next step: subtraction.

Subtracting the Fractions

With both fractions now sharing a common denominator of 20, we can proceed with the subtraction. The process of subtracting fractions with a common denominator is straightforward: we simply subtract the numerators and keep the denominator the same. In our case, we have ${\frac{15}{20}}$ and ${\frac{8}{20}}$, so we subtract 8 from 15:

${\frac{15}{20} - \frac{8}{20} = \frac{15 - 8}{20} = \frac{7}{20}}$

Therefore, the result of subtracting ${\frac{8}{20}}$ from ${\frac{15}{20}}$ is ${\frac{7}{20}}$. This fraction represents the difference between the two original fractions, ${\frac{3}{4}}$ and ${\frac{2}{5}}$. It's important to remember that we only subtract the numerators; the denominator remains the same because it represents the size of the parts we are working with. The result, ${\frac{7}{20}}$, is a fraction in its simplest form, as 7 and 20 share no common factors other than 1.

Simplifying the Result (If Necessary)

After performing the subtraction, it's essential to check if the resulting fraction can be simplified. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. In other words, we want to ensure that the fraction is expressed in its lowest terms. To simplify a fraction, we look for the greatest common factor (GCF) of the numerator and the denominator and then divide both by that factor.

In our case, the result of the subtraction is ${\frac{7}{20}}$. The numerator is 7, and the denominator is 20. The factors of 7 are 1 and 7, and the factors of 20 are 1, 2, 4, 5, 10, and 20. The only common factor between 7 and 20 is 1. Since the GCF is 1, the fraction ${\frac{7}{20}}$ is already in its simplest form. There is no further simplification required. However, if we had obtained a fraction like ${\frac{10}{20}}$, we would simplify it by dividing both the numerator and the denominator by their GCF, which is 10, resulting in ${\frac{1}{2}}$. Checking for simplification ensures that we present the answer in the most concise and understandable form.

The Final Answer

Having followed all the necessary steps, we arrive at the final answer. We started with the problem ${\frac{3}{4} - \frac{2}{5}}$ and systematically worked through the process of finding a common denominator, creating equivalent fractions, subtracting the numerators, and simplifying the result. The final result of the subtraction is ${\frac{7}{20}}$. This fraction represents the difference between ${\frac{3}{4}}$ and ${\frac{2}{5}}$. It's crucial to clearly state the final answer to ensure the solution is complete and easily understood. The journey to this answer involved several key concepts in fraction arithmetic, highlighting the importance of understanding each step. From identifying the need for a common denominator to simplifying the final result, each stage contributes to a solid understanding of fraction subtraction.

Real-World Applications

Understanding fraction subtraction isn't just about solving problems in a textbook; it has numerous real-world applications. From cooking and baking to measuring and construction, fractions are an integral part of everyday life. For instance, imagine you have ${\frac{3}{4}}$ of a cup of flour and a recipe calls for ${\frac{2}{5}}$ of a cup. To determine how much flour you'll have left after making the recipe, you need to subtract ${\frac{2}{5}}$ from ${\frac{3}{4}}$, which, as we've learned, equals ${\frac{7}{20}}$ of a cup. Similarly, in construction, fractions are used to measure lengths and distances. If you need to cut a piece of wood that is ${\frac{2}{5}}$ of a meter shorter than a piece that is ${\frac{3}{4}}$ of a meter long, you'll need to perform this subtraction. These examples illustrate how the ability to subtract fractions is a practical skill that can be applied in various situations, making it an essential part of mathematical literacy.

Conclusion: Mastering Fraction Subtraction

In conclusion, mastering fraction subtraction is a crucial skill in mathematics with wide-ranging applications. We've thoroughly explored the process of solving the problem ${\frac{3}{4} - \frac{2}{5}}$, breaking it down into manageable steps: identifying the need for a common denominator, finding the Least Common Denominator (LCD), creating equivalent fractions, subtracting the numerators, and simplifying the result. The final answer, ${\frac{7}{20}}$, represents the difference between the two fractions. This step-by-step approach not only provides a solution to this specific problem but also equips you with the knowledge and confidence to tackle other fraction subtraction challenges. The real-world examples highlight the practical importance of this skill, emphasizing its relevance in everyday situations. By understanding the underlying principles and practicing regularly, you can achieve mastery in fraction subtraction and unlock further mathematical concepts.