Mastering Fraction And Decimal Arithmetic Step-by-Step Solutions

Let's dive deep into this multifaceted fraction arithmetic problem. Fractions, fractions, fractions, they might seem daunting at first, but with a systematic approach, we can conquer them. To break down this equation, we'll address each operation step-by-step, ensuring a clear and comprehensive understanding. First, we'll tackle the addition and subtraction of fractions, which is the cornerstone of this problem. When adding or subtracting fractions, it's crucial to have a common denominator. In this case, all fractions conveniently share the denominator of 5, simplifying our initial calculations. So, let's begin by focusing on the first part of the equation: (3/5) + (2/5) - (5+7)/5. We'll first deal with the addition: (3/5) + (2/5). Since they have a common denominator, we simply add the numerators: 3 + 2 = 5. This gives us 5/5, which simplifies to 1. Now, we move on to the subtraction. The equation now looks like this: 1 - (5+7)/5. Next, we need to simplify the term (5+7)/5. First, we add 5 and 7, which equals 12. This gives us 12/5. So, the equation becomes 1 - 12/5. To subtract a fraction from a whole number, we need to convert the whole number into a fraction with the same denominator as the fraction we're subtracting. In this case, we convert 1 into 5/5. Now we have 5/5 - 12/5. Subtracting the numerators, we get 5 - 12 = -7. Thus, the result of the subtraction is -7/5. Next, we address the division part of the equation: (12/5) ÷ 2. To divide a fraction by a whole number, we can think of the whole number as a fraction with a denominator of 1. So, we have (12/5) ÷ (2/1). Dividing fractions involves multiplying by the reciprocal of the divisor. The reciprocal of 2/1 is 1/2. Therefore, we multiply 12/5 by 1/2: (12/5) * (1/2). Multiplying the numerators, we get 12 * 1 = 12. Multiplying the denominators, we get 5 * 2 = 10. This gives us 12/10. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2. Dividing 12 by 2 gives us 6, and dividing 10 by 2 gives us 5. So, the simplified fraction is 6/5. Finally, let's compare the two sides of the original equation: -7/5 = 6/5. This statement is clearly false. Therefore, there might be a misunderstanding or a mistake in the original equation. It's crucial to double-check the equation and the steps taken to solve it. In conclusion, by systematically breaking down the equation and applying the rules of fraction arithmetic, we can navigate complex problems with confidence. Remember, practice makes perfect, and the more you work with fractions, the more comfortable you'll become with them.

Moving on to the next problem, we encounter the addition of two fractions with different denominators. Adding fractions requires a common denominator, and this is where the concept of finding the least common multiple (LCM) comes into play. Our fractions are (1/4) and (3/8). The denominators are 4 and 8. To find the LCM, we look for the smallest number that both 4 and 8 can divide into evenly. In this case, the LCM of 4 and 8 is 8. Now, we need to convert the fraction (1/4) into an equivalent fraction with a denominator of 8. To do this, we multiply both the numerator and the denominator of (1/4) by the same number so that the denominator becomes 8. Since 4 * 2 = 8, we multiply both the numerator and the denominator by 2: (1 * 2) / (4 * 2) = 2/8. So, (1/4) is equivalent to (2/8). Now, we can rewrite the original equation as (2/8) + (3/8). Since both fractions now have the same denominator, we can add the numerators directly: 2 + 3 = 5. This gives us 5/8. The fraction 5/8 is already in its simplest form because 5 and 8 have no common factors other than 1. Therefore, the final answer is 5/8. This problem highlights the importance of finding a common denominator when adding fractions. It's a fundamental concept in fraction arithmetic, and mastering it is crucial for tackling more complex problems. Remember, when working with fractions, always simplify your answers to their lowest terms. This involves dividing both the numerator and the denominator by their greatest common factor (GCF). This ensures that your answer is in the most concise and understandable form. In this specific case, the fraction 5/8 is already in its simplest form, but in other cases, simplification might be necessary. Understanding the relationship between fractions and their equivalent forms is also crucial. For example, we saw that (1/4) is equivalent to (2/8). Recognizing these equivalencies allows us to manipulate fractions and solve problems more effectively. In summary, adding fractions with different denominators involves finding a common denominator, converting the fractions to equivalent forms with the common denominator, adding the numerators, and simplifying the result if necessary. This process is a cornerstone of fraction arithmetic and a valuable skill in mathematics.

Here we have another fraction addition problem, emphasizing the crucial step of finding a common denominator. Fraction addition becomes straightforward once the denominators are aligned. Our equation is (2/3) + (1/6). The denominators are 3 and 6. To add these fractions, we need to find the least common multiple (LCM) of 3 and 6. The LCM is the smallest number that both 3 and 6 divide into evenly. In this case, the LCM of 3 and 6 is 6. Now, we need to convert the fraction (2/3) into an equivalent fraction with a denominator of 6. To do this, we multiply both the numerator and the denominator of (2/3) by the same number so that the denominator becomes 6. Since 3 * 2 = 6, we multiply both the numerator and the denominator by 2: (2 * 2) / (3 * 2) = 4/6. So, (2/3) is equivalent to (4/6). Now, we can rewrite the original equation as (4/6) + (1/6). Since both fractions now have the same denominator, we can add the numerators directly: 4 + 1 = 5. This gives us 5/6. The fraction 5/6 is already in its simplest form because 5 and 6 have no common factors other than 1. Therefore, the final answer is 5/6. This problem reinforces the importance of finding a common denominator when adding fractions. It's a fundamental concept in fraction arithmetic, and mastering it is crucial for tackling more complex problems. Remember, when working with fractions, always check if your answer can be simplified. Simplification involves dividing both the numerator and the denominator by their greatest common factor (GCF). This ensures that your answer is in the most concise and understandable form. In this specific case, the fraction 5/6 is already in its simplest form, but in other cases, simplification might be necessary. In summary, adding fractions with different denominators involves finding a common denominator, converting the fractions to equivalent forms with the common denominator, adding the numerators, and simplifying the result if necessary. This process is a cornerstone of fraction arithmetic and a valuable skill in mathematics. Understanding equivalent fractions and how to find the LCM are key to mastering fraction addition.

Now, let's shift our focus to decimal addition. Decimal addition is similar to adding whole numbers, but it's crucial to align the decimal points correctly. In this problem, we have 0.4 + 0.6. To add these decimals, we can stack them vertically, ensuring that the decimal points are aligned: 0. 4 + 0. 6 ------ Now, we add the digits in each column, starting from the rightmost column. In the tenths column, we have 4 + 6 = 10. We write down the 0 and carry over the 1 to the ones column. In the ones column, we have 0 + 0 + 1 (carried over) = 1. So, the sum is 1.0, which is simply 1. Therefore, 0.4 + 0.6 = 1. This problem illustrates the fundamental principle of decimal addition: aligning the decimal points. When the decimal points are aligned, we are essentially adding digits with the same place value (tenths, hundredths, etc.). This ensures that we get the correct result. Another way to think about this problem is to convert the decimals into fractions. 0.4 is equivalent to 4/10, and 0.6 is equivalent to 6/10. Adding these fractions, we get 4/10 + 6/10. Since they have a common denominator, we can add the numerators: 4 + 6 = 10. This gives us 10/10, which simplifies to 1. This approach reinforces the connection between decimals and fractions and provides an alternative way to solve the problem. In summary, decimal addition involves aligning the decimal points, adding the digits in each column, and carrying over when necessary. It's a fundamental arithmetic skill with applications in various real-world scenarios. Whether you prefer adding decimals directly or converting them to fractions, understanding the underlying principles is key to success.

Finally, let's tackle another decimal addition problem. This example will further solidify our understanding of decimal arithmetic. We are presented with the equation: 2.75 + 1.25. As with the previous decimal addition problem, the key here is to align the decimal points correctly. Let's stack the numbers vertically: 2. 75 + 1. 25 ------ Now, we add the digits in each column, starting from the rightmost column (the hundredths column). In the hundredths column, we have 5 + 5 = 10. We write down the 0 and carry over the 1 to the tenths column. In the tenths column, we have 7 + 2 + 1 (carried over) = 10. We write down the 0 and carry over the 1 to the ones column. In the ones column, we have 2 + 1 + 1 (carried over) = 4. So, the sum is 4.00, which simplifies to 4. Therefore, 2.75 + 1.25 = 4. This problem further illustrates the importance of aligning decimal points when adding decimals. By aligning the decimal points, we ensure that we are adding digits with the same place value. This is crucial for obtaining the correct sum. We can also approach this problem by thinking about the place values of the digits. In the number 2.75, the 2 represents 2 ones, the 7 represents 7 tenths (or 70 hundredths), and the 5 represents 5 hundredths. Similarly, in the number 1.25, the 1 represents 1 one, the 2 represents 2 tenths (or 20 hundredths), and the 5 represents 5 hundredths. Adding the ones, we get 2 + 1 = 3. Adding the tenths, we get 0.7 + 0.2 = 0.9. Adding the hundredths, we get 0.05 + 0.05 = 0.10. Combining these, we have 3 + 0.9 + 0.10 = 4. This method provides a deeper understanding of the place value system and how it relates to decimal addition. In summary, decimal addition involves aligning the decimal points, adding the digits in each column, and carrying over when necessary. It's a fundamental arithmetic skill that is essential for various mathematical and real-world applications. Practice and a solid understanding of place value will help you master decimal addition.

In conclusion, mastering fraction and decimal arithmetic involves understanding the underlying principles, practicing regularly, and applying systematic approaches to problem-solving. Whether it's finding common denominators for fraction addition or aligning decimal points for decimal addition, these skills are crucial for mathematical proficiency. Keep practicing, and you'll become a master of arithmetic!