Solving (+1/6)-(-1/2) A Step-by-Step Math Guide

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Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of fractions and wondered where to even begin? Well, you're not alone! Fractions can seem intimidating at first, but with a little bit of understanding and some simple steps, you'll be subtracting them like a pro in no time. In this article, we're going to break down the problem (+1/6)-(-1/2) and show you exactly how to solve it. We'll explore the underlying concepts, walk through each step in detail, and even throw in some helpful tips and tricks along the way. So, grab your pencils, notebooks, and let's dive into the world of fraction subtraction!

Fractions, at their core, represent parts of a whole. Think of a pizza cut into slices – each slice is a fraction of the entire pizza. A fraction consists of two main parts: the numerator (the top number) and the denominator (the bottom number). The numerator tells us how many parts we have, while the denominator tells us the total number of parts the whole is divided into. For example, in the fraction 1/2, the numerator is 1, and the denominator is 2. This means we have one part out of a total of two parts – essentially, half of the whole. Similarly, 1/6 represents one part out of a total of six parts.

Now, when it comes to subtracting fractions, things get a little more interesting. We can only directly subtract fractions if they have the same denominator, which is known as a common denominator. This makes sense if you think about it in terms of our pizza analogy. You can't easily subtract slices from two pizzas cut into different numbers of slices. You need to have the slices represent the same fraction of the whole. So, to subtract fractions with different denominators, we need to find a common denominator first. This involves finding a common multiple of the denominators, which is a number that both denominators divide into evenly. The least common multiple (LCM) is the smallest such number, and it's often the easiest common denominator to work with. Once we have a common denominator, we can rewrite the fractions with this new denominator and then subtract the numerators. The denominator stays the same.

In our problem, (+1/6)-(-1/2), we're dealing with two fractions: 1/6 and -1/2. Notice that one of the fractions is negative. Don't let that scare you! Subtracting a negative number is the same as adding the positive version of that number. This is a crucial concept to remember when working with negative fractions. So, the problem (+1/6)-(-1/2) can be rewritten as (+1/6) + (+1/2). Now, we have a fraction addition problem. But before we can add these fractions, we need to find a common denominator. Take a look at the denominators, 6 and 2. What's the smallest number that both 6 and 2 divide into evenly? That's right, it's 6! So, 6 will be our common denominator.

Step-by-Step Solution: Cracking the Code of (+1/6)-(-1/2)

Alright, let's get down to business and solve this problem step-by-step. We've already laid the groundwork by understanding the basics of fractions and common denominators. Now, it's time to put those concepts into action and see how it all comes together.

Step 1: Rewrite the Problem

As we discussed earlier, subtracting a negative number is the same as adding its positive counterpart. So, let's rewrite our problem to make it a bit easier to work with:

(+1/6) - (-1/2) becomes (+1/6) + (+1/2)

This simple change makes the problem look less intimidating and sets us up for the next step.

Step 2: Find the Common Denominator

Now, we need to find a common denominator for the fractions 1/6 and 1/2. Remember, a common denominator is a number that both denominators divide into evenly. In this case, we're looking for the least common multiple (LCM) of 6 and 2.

  • Multiples of 6: 6, 12, 18, 24...
  • Multiples of 2: 2, 4, 6, 8, 10...

The smallest number that appears in both lists is 6. So, 6 is our least common multiple and our common denominator.

Step 3: Rewrite the Fractions with the Common Denominator

Now, we need to rewrite each fraction with a denominator of 6. The fraction 1/6 already has a denominator of 6, so we don't need to change it. But the fraction 1/2 needs to be adjusted.

To rewrite 1/2 with a denominator of 6, we need to multiply both the numerator and the denominator by the same number. What number do we multiply 2 by to get 6? That's right, it's 3.

So, we multiply both the numerator and denominator of 1/2 by 3:

(1 * 3) / (2 * 3) = 3/6

Now we have two fractions with the same denominator: 1/6 and 3/6.

Step 4: Add the Fractions

With a common denominator in place, we can now add the fractions. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same.

1/6 + 3/6 = (1 + 3) / 6 = 4/6

So, the sum of the fractions is 4/6.

Step 5: Simplify the Fraction (if possible)

Our final answer is 4/6, but we can simplify this fraction further. Simplifying a fraction means reducing it to its lowest terms. To do this, we need to find the greatest common factor (GCF) of the numerator and denominator and divide both by it.

The factors of 4 are: 1, 2, 4 The factors of 6 are: 1, 2, 3, 6

The greatest common factor of 4 and 6 is 2.

So, we divide both the numerator and denominator of 4/6 by 2:

(4 / 2) / (6 / 2) = 2/3

Therefore, the simplified fraction is 2/3.

The Final Answer: Unveiling the Solution to (+1/6)-(-1/2)

Drumroll, please! After all those steps, we've finally arrived at the answer. The solution to the problem (+1/6)-(-1/2) is:

2/3

Congratulations! You've successfully navigated the world of fraction subtraction and conquered this problem. You've not only found the answer but also gained a deeper understanding of the concepts involved. But hold on, the journey doesn't end here. Let's explore some more tips and tricks to solidify your understanding and tackle even more complex fraction problems.

Pro Tips and Tricks: Mastering the Art of Fraction Subtraction

Now that you've mastered the basics of fraction subtraction, let's take things up a notch with some pro tips and tricks. These insights will help you solve problems more efficiently and confidently, and even impress your friends with your fraction skills!

Tip 1: Visualizing Fractions

Sometimes, the best way to understand fractions is to visualize them. Imagine a pie cut into equal slices. Each slice represents a fraction of the whole pie. For example, 1/2 is one slice out of two, 1/4 is one slice out of four, and so on. Visualizing fractions can help you grasp the concept of common denominators and how adding or subtracting fractions changes the size of the pie slices you have.

You can even draw diagrams to represent fractions. Draw a rectangle and divide it into the number of parts indicated by the denominator. Then, shade in the number of parts indicated by the numerator. This visual representation can make it easier to compare fractions, find common denominators, and perform operations.

Tip 2: Using the Butterfly Method

The butterfly method is a handy shortcut for adding or subtracting two fractions. It's a visual technique that can help you find the common denominator and perform the necessary calculations quickly.

Here's how it works:

  1. Draw a butterfly shape connecting the numerators and denominators of the two fractions.
  2. Multiply the numerator of the first fraction by the denominator of the second fraction. This gives you one "wing" of the butterfly.
  3. Multiply the numerator of the second fraction by the denominator of the first fraction. This gives you the other "wing" of the butterfly.
  4. Add or subtract the wings, depending on whether you're adding or subtracting the fractions. This gives you the numerator of the result.
  5. Multiply the two denominators together. This gives you the denominator of the result.

For example, let's use the butterfly method to solve 1/6 + 1/2:

  1. Draw a butterfly shape connecting 1 and 2, and 1 and 6.
  2. 1 * 2 = 2 (one wing)
  3. 1 * 6 = 6 (the other wing)
  4. 2 + 6 = 8 (numerator of the result)
  5. 6 * 2 = 12 (denominator of the result)

So, 1/6 + 1/2 = 8/12. Now, simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 4. This gives you 2/3, the same answer we got using the step-by-step method.

Tip 3: Dealing with Mixed Numbers

Sometimes, you'll encounter mixed numbers, which are numbers that combine a whole number and a fraction (e.g., 1 1/2). To add or subtract mixed numbers, you have two options:

  1. Convert to Improper Fractions: Turn each mixed number into an improper fraction (where the numerator is greater than the denominator) and then perform the addition or subtraction as usual. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator.
  2. Add/Subtract Whole Numbers and Fractions Separately: Add or subtract the whole number parts and the fractional parts separately. If the fractional parts result in an improper fraction, convert it to a mixed number and add the whole number part to the whole number sum.

Choose the method that feels most comfortable and efficient for you.

Tip 4: Practice Makes Perfect

Like any skill, mastering fraction subtraction takes practice. The more problems you solve, the more comfortable you'll become with the concepts and techniques. So, don't be afraid to tackle a variety of fraction problems, from simple ones to more complex ones. You can find plenty of practice problems in textbooks, online resources, and even worksheets. The key is to keep practicing and challenging yourself.

Conclusion: You've Conquered the Fraction Challenge!

Well, there you have it! You've successfully navigated the world of fraction subtraction and learned how to solve the problem (+1/6)-(-1/2). You've explored the fundamental concepts, mastered the step-by-step solution, and even picked up some pro tips and tricks along the way. Give yourself a pat on the back – you've earned it!

Remember, fractions might seem daunting at first, but with a little bit of understanding and consistent practice, you can conquer any fraction challenge that comes your way. So, keep practicing, keep exploring, and keep unlocking the fascinating world of mathematics. And who knows, maybe you'll even start to enjoy working with fractions!