Solving Fraction Arithmetic A Step-by-Step Guide To (25/5)-(35/25)+(8/35)-(4/60)

Introduction

Fraction arithmetic can often seem daunting, but with a systematic approach and a clear understanding of the underlying principles, complex expressions can be simplified and solved with ease. This article delves into the step-by-step solution of the expression ${\frac{25}{5}-\frac{35}{25}+\frac{8}{35}-\frac{4}{60}}$, providing a comprehensive guide suitable for students, educators, and anyone looking to brush up on their math skills. Understanding how to manipulate fractions, find common denominators, and reduce fractions to their simplest form is crucial in various fields, including mathematics, engineering, and everyday problem-solving. By breaking down each step and explaining the rationale behind it, we aim to make this process accessible and understandable to all.

This guide not only provides the solution but also offers insights into the strategies and techniques used. We will explore the importance of simplifying fractions, finding the least common multiple (LCM) to achieve common denominators, and performing the arithmetic operations accurately. The goal is to empower readers with the knowledge and skills to tackle similar problems confidently. Whether you are a student preparing for an exam or someone looking to enhance your mathematical proficiency, this article serves as a valuable resource. We will address common pitfalls and offer tips to avoid mistakes, ensuring a thorough grasp of the concepts. So, let's embark on this mathematical journey and demystify the world of fraction arithmetic together.

Step 1: Simplifying the Fractions

The initial step in solving any fractional expression is to simplify each fraction individually. This not only makes the subsequent calculations easier but also reduces the chances of errors. Simplification involves dividing both the numerator and the denominator of a fraction by their greatest common divisor (GCD). This process reduces the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. In our expression, we have four fractions: ${\frac{25}{5}}$, ${\frac{35}{25}}$, ${\frac{8}{35}}$, and ${\frac{4}{60}}$. Let's simplify each one.

Starting with ${\frac{25}{5}}$, we can see that both 25 and 5 are divisible by 5. Dividing both the numerator and the denominator by 5, we get ${\frac{25 ÷ 5}{5 ÷ 5} = \frac{5}{1}}$, which simplifies to 5. This simplification significantly reduces the complexity of the expression. Next, consider ${\frac{35}{25}}$. Both 35 and 25 are divisible by 5. Dividing both by 5 gives us ${\frac{35 ÷ 5}{25 ÷ 5} = \frac{7}{5}}$. This fraction is now in its simplest form as 7 and 5 have no common factors other than 1. Moving on to ${\frac{8}{35}}$, we need to find the GCD of 8 and 35. The factors of 8 are 1, 2, 4, and 8, while the factors of 35 are 1, 5, 7, and 35. The only common factor is 1, so ${\frac{8}{35}}$ is already in its simplest form. Finally, let's simplify ${\frac{4}{60}}$. Both 4 and 60 are divisible by 4. Dividing both by 4 gives us ${\frac{4 ÷ 4}{60 ÷ 4} = \frac{1}{15}}$. Now that we have simplified each fraction, the expression becomes much more manageable: ${5 - \frac{7}{5} + \frac{8}{35} - \frac{1}{15}}$. This simplification is a crucial step in making the subsequent calculations more straightforward and less prone to errors.

Step 2: Finding the Least Common Denominator (LCD)

After simplifying the fractions, the next crucial step is to find the least common denominator (LCD). The LCD is the smallest multiple that is common to all the denominators in the expression. Finding the LCD is essential because we need a common denominator to perform addition and subtraction of fractions. In our simplified expression, we have the following denominators: 1 (from the simplified 5, which is ${\frac{5}{1}}$), 5, 35, and 15. To find the LCD, we need to determine the least common multiple (LCM) of these numbers. There are several methods to find the LCM, such as listing multiples or using prime factorization.

Let’s use the prime factorization method. First, we find the prime factorization of each denominator: 1 = 1, 5 = 5, 35 = 5 × 7, and 15 = 3 × 5. Now, we identify the highest power of each prime factor that appears in any of the factorizations. We have the prime factors 3, 5, and 7. The highest power of 3 is 3¹ (from 15), the highest power of 5 is 5¹ (from 5, 35, and 15), and the highest power of 7 is 7¹ (from 35). To find the LCM, we multiply these highest powers together: LCM = 3¹ × 5¹ × 7¹ = 3 × 5 × 7 = 105. Therefore, the least common denominator (LCD) for our expression is 105. Now that we have the LCD, we can proceed to convert each fraction to an equivalent fraction with the denominator of 105. This involves multiplying both the numerator and the denominator of each fraction by the factor that will make the denominator equal to 105. This step is critical for accurately adding and subtracting the fractions. Once all fractions have the common denominator, we can combine them and proceed with the arithmetic operations.

Step 3: Converting Fractions to Equivalent Fractions with the LCD

Now that we've found the least common denominator (LCD) to be 105, the next step is to convert each fraction into an equivalent fraction with this common denominator. This process ensures that we can perform addition and subtraction accurately. To convert a fraction to an equivalent fraction with the LCD, we need to multiply both the numerator and the denominator by the same factor. This factor is determined by dividing the LCD by the original denominator of the fraction. Let's apply this process to each term in our expression: ${5 - \frac{7}{5} + \frac{8}{35} - \frac{1}{15}}$.

First, we convert 5, which can be written as ${\frac{5}{1}}$. To get the denominator to 105, we multiply both the numerator and the denominator by 105: ${\frac{5 × 105}{1 × 105} = \frac{525}{105}}$. Next, we convert ${\frac{7}{5}}$. To get the denominator to 105, we divide 105 by 5, which gives us 21. So, we multiply both the numerator and the denominator by 21: ${\frac{7 × 21}{5 × 21} = \frac{147}{105}}$. Moving on to ${\frac{8}{35}}$, we divide 105 by 35, which gives us 3. We multiply both the numerator and the denominator by 3: ${\frac{8 × 3}{35 × 3} = \frac{24}{105}}$. Finally, we convert ${\frac{1}{15}}$. We divide 105 by 15, which gives us 7. We multiply both the numerator and the denominator by 7: ${\frac{1 × 7}{15 × 7} = \frac{7}{105}}$. Now, our expression looks like this: ${\frac{525}{105} - \frac{147}{105} + \frac{24}{105} - \frac{7}{105}}$. Each fraction now has the same denominator, making it possible to perform the addition and subtraction operations in the next step. This conversion is a critical step in solving the expression accurately.

Step 4: Performing the Arithmetic Operations

With all fractions now having a common denominator of 105, we can proceed to perform the arithmetic operations. This involves adding and subtracting the numerators while keeping the denominator constant. Our expression is now ${\frac{525}{105} - \frac{147}{105} + \frac{24}{105} - \frac{7}{105}}$. To solve this, we perform the operations from left to right. First, we subtract ${\frac{147}{105}}$ from ${\frac{525}{105}}$: ${\frac{525 - 147}{105} = \frac{378}{105}}$. Next, we add ${\frac{24}{105}}$ to the result: ${\frac{378 + 24}{105} = \frac{402}{105}}$. Finally, we subtract ${\frac{7}{105}}$ from the result: ${\frac{402 - 7}{105} = \frac{395}{105}}$. So, the result of the arithmetic operations is ${\frac{395}{105}}$. However, this is not the final answer, as the fraction can be simplified further. Performing the arithmetic operations with a common denominator makes the calculations straightforward. The key is to ensure that each step is performed accurately, and the numerators are combined correctly. Once we have the result, the next step is to simplify the fraction to its lowest terms, which will give us the final answer.

Step 5: Simplifying the Final Fraction

After performing the arithmetic operations, we arrived at the fraction ${\frac{395}{105}}$. The final step in solving the expression is to simplify this fraction to its lowest terms. Simplifying a fraction involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by this GCD. This process reduces the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. To find the GCD of 395 and 105, we can use methods such as listing factors or using the Euclidean algorithm.

Let's use the prime factorization method. First, we find the prime factorization of 395 and 105. The prime factorization of 395 is 5 × 79, and the prime factorization of 105 is 3 × 5 × 7. The common prime factor between 395 and 105 is 5. Therefore, the GCD of 395 and 105 is 5. Now, we divide both the numerator and the denominator by the GCD: ${\frac{395 ÷ 5}{105 ÷ 5} = \frac{79}{21}}$. The fraction ${\frac{79}{21}}$ is now in its simplest form because 79 is a prime number, and it does not share any common factors with 21 other than 1. So, the simplified fraction is ${\frac{79}{21}}$. This fraction can also be expressed as a mixed number. To convert ${\frac{79}{21}}$ to a mixed number, we divide 79 by 21. The quotient is 3, and the remainder is 16. Therefore, the mixed number is 3 ${\frac{16}{21}}$. Simplifying the final fraction is a crucial step as it presents the answer in its most concise and understandable form. In this case, the final simplified answer is ${\frac{79}{21}}$ or 3 ${\frac{16}{21}}$.

Conclusion

In conclusion, solving the expression ${\frac{25}{5}-\frac{35}{25}+\frac{8}{35}-\frac{4}{60}}$ involves a series of steps that, when followed methodically, lead to a clear and accurate solution. We began by simplifying each fraction, which made the subsequent calculations more manageable. Next, we found the least common denominator (LCD), a critical step for adding and subtracting fractions. We then converted each fraction to an equivalent fraction with the LCD, ensuring that all terms were compatible for arithmetic operations. After this, we performed the arithmetic operations, combining the numerators while keeping the common denominator. Finally, we simplified the resulting fraction to its lowest terms, arriving at the final answer of ${\frac{79}{21}}$ or 3 ${\frac{16}{21}}$.

This process highlights the importance of understanding fundamental mathematical principles and applying them systematically. Each step, from simplifying fractions to finding the LCD and performing arithmetic operations, builds upon the previous one, leading to a coherent and accurate solution. The ability to manipulate fractions is a valuable skill, not only in mathematics but also in various real-world applications. By breaking down the problem into smaller, manageable steps, we have demonstrated how complex expressions can be tackled with confidence. The key takeaway is that with a clear understanding of the rules and a methodical approach, anyone can master fraction arithmetic and solve similar problems effectively.