Solving (3/2-1/2)+(1/3-1/5) A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of fraction operations, specifically tackling the expression (3/2 - 1/2) + (1/3 - 1/5). This might seem a bit daunting at first, but don't worry! We're going to break it down step-by-step, ensuring you not only understand the solution but also grasp the underlying concepts. So, grab your pencils and notebooks, and let's get started on this fractional adventure!

Understanding the Basics of Fraction Operations

Before we jump into solving our main expression, let's quickly refresh the fundamental concepts of fraction operations. Fractions, as we know, represent parts of a whole. The top number in a fraction is called the numerator, and it indicates how many parts we have. The bottom number is the denominator, which tells us the total number of equal parts the whole is divided into.

When we perform operations like addition and subtraction with fractions, things get a little interesting. The most crucial thing to remember is that you can only add or subtract fractions directly if they have the same denominator. Think of it like trying to add apples and oranges – they're different units! To add or subtract, you need to find a common denominator, which is a shared multiple of the denominators involved. This is where the concept of the Least Common Multiple (LCM) comes into play. The LCM is the smallest number that is a multiple of two or more numbers. Finding the LCM helps us convert fractions to equivalent forms with the same denominator, making the addition or subtraction process smooth and accurate.

Moreover, understanding equivalent fractions is paramount. Equivalent fractions represent the same value but have different numerators and denominators. For example, 1/2 and 2/4 are equivalent fractions. To create an equivalent fraction, you multiply (or divide) both the numerator and the denominator by the same non-zero number. This principle is key when we need to convert fractions to a common denominator. Remember, our goal is to manipulate the fractions so they can be easily added or subtracted. Mastering these basic concepts will make dealing with more complex expressions, like the one we're tackling today, much more manageable. So, let's keep these fundamentals in mind as we move forward, and we'll see how they come into play in solving (3/2 - 1/2) + (1/3 - 1/5).

Step-by-Step Solution for (3/2 - 1/2) + (1/3 - 1/5)

Okay, let's dive into solving the expression (3/2 - 1/2) + (1/3 - 1/5) step-by-step. The golden rule when dealing with mathematical expressions is to follow the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). In our case, we have parentheses, so we'll tackle the operations within them first.

First, let's focus on the first set of parentheses: (3/2 - 1/2). Here, we have two fractions with the same denominator, which makes our lives much easier! When the denominators are the same, we can simply subtract the numerators and keep the denominator as is. So, 3/2 - 1/2 becomes (3 - 1) / 2, which simplifies to 2/2. Now, 2/2 is a fraction that can be further simplified because both the numerator and the denominator are the same. A fraction where the numerator and denominator are equal is equivalent to 1. Therefore, (3/2 - 1/2) = 1.

Next, we'll tackle the second set of parentheses: (1/3 - 1/5). Here, we encounter fractions with different denominators, so we need to find a common denominator before we can subtract. To do this, we need to find the Least Common Multiple (LCM) of 3 and 5. The LCM of 3 and 5 is 15. Now, we'll convert both fractions to equivalent fractions with a denominator of 15. To convert 1/3 to an equivalent fraction with a denominator of 15, we multiply both the numerator and the denominator by 5 (since 3 * 5 = 15). This gives us (1 * 5) / (3 * 5) = 5/15. Similarly, to convert 1/5 to an equivalent fraction with a denominator of 15, we multiply both the numerator and the denominator by 3 (since 5 * 3 = 15). This gives us (1 * 3) / (5 * 3) = 3/15. Now we can subtract: 5/15 - 3/15. Subtracting the numerators gives us (5 - 3) / 15, which simplifies to 2/15. So, (1/3 - 1/5) = 2/15.

Finally, we add the results from both sets of parentheses: 1 + 2/15. To add a whole number to a fraction, we can think of the whole number as a fraction with a denominator of 1. So, 1 becomes 1/1. Now we need a common denominator to add 1/1 and 2/15. The LCM of 1 and 15 is 15. Converting 1/1 to an equivalent fraction with a denominator of 15, we multiply both the numerator and the denominator by 15, which gives us 15/15. Now we can add: 15/15 + 2/15. Adding the numerators gives us (15 + 2) / 15, which simplifies to 17/15. Therefore, the final result of the expression (3/2 - 1/2) + (1/3 - 1/5) is 17/15. This can also be expressed as a mixed number: 1 and 2/15. And there you have it! We've successfully navigated through the fraction operations and arrived at our solution.

Common Mistakes and How to Avoid Them

Now that we've solved the expression (3/2 - 1/2) + (1/3 - 1/5), let's talk about some common pitfalls people encounter when working with fractions and how to avoid them. Recognizing these mistakes can save you a lot of frustration and ensure accuracy in your calculations.

One of the most frequent errors is attempting to add or subtract fractions without finding a common denominator first. Remember, fractions need to represent parts of the same whole to be directly added or subtracted. It's like trying to add apples and oranges – you need a common unit (like