Fractions Made Easy: Solve 2/3 - 1/2 + 8/15 Step-by-Step

by Sebastian Müller 57 views

Hey guys! Ever stumbled upon a fraction problem that looked like a mathematical maze? Don't worry; we've all been there. Fractions can seem intimidating, but with a systematic approach and a sprinkle of practice, they become as easy as pie. In this article, we're going to break down a specific problem: 2/3 - 1/2 + 8/15. We'll walk through each step, ensuring you not only get the answer but also understand the why behind it. So, grab your pencils, and let's dive into the world of fractions!

Understanding the Basics: Why Common Denominators Matter

Before we jump into the solution, let's quickly recap the fundamental principle behind adding and subtracting fractions. Imagine you have a pizza cut into different sized slices. You can't easily say how much pizza you have if some slices are large and others are tiny, right? You need to cut the pizza into equal-sized slices first. That's where the concept of a common denominator comes in.

In fraction arithmetic, the denominator represents the total number of equal parts a whole is divided into. To add or subtract fractions, these parts need to be the same size – hence, the need for a common denominator. It's like making sure all our pizza slices are the same size before counting them. So, before we can tackle 2/3 - 1/2 + 8/15, we need to find a common denominator for 3, 2, and 15. But how do we find this magical number?

Finding the Least Common Multiple (LCM): The easiest way to find a common denominator is to determine the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of all the denominators. There are a couple of ways to find the LCM. One way is to list out the multiples of each number until you find a common one. For example:

  • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
  • Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, ...
  • Multiples of 15: 15, 30, 45, ...

As you can see, 30 is a common multiple of 3, 2, and 15. However, it might not be the least common multiple. Another approach is to use prime factorization. This method is particularly useful when dealing with larger numbers. Let's break down 3, 2, and 15 into their prime factors:

  • 3 = 3
  • 2 = 2
  • 15 = 3 x 5

To find the LCM, we take the highest power of each prime factor that appears in any of the numbers. In this case, we have 2, 3, and 5 as prime factors. The highest power of 2 is 2¹ (from the number 2), the highest power of 3 is 3¹ (from the numbers 3 and 15), and the highest power of 5 is 5¹ (from the number 15). Multiplying these together, we get:

LCM = 2¹ x 3¹ x 5¹ = 2 x 3 x 5 = 30

So, the Least Common Multiple (LCM) of 3, 2, and 15 is 30. This means 30 will be our common denominator. Now that we have our common denominator, we can move on to the next step: converting the fractions.

Step-by-Step Solution: Conquering the Fraction Problem

Now that we've found our common denominator, 30, let's rewrite each fraction with this new denominator. This involves figuring out what to multiply the original denominator by to get 30, and then multiplying the numerator by the same number. Remember, we need to keep the value of the fraction the same, so we're essentially multiplying each fraction by a clever form of 1.

1. Convert 2/3 to an equivalent fraction with a denominator of 30:

To get from 3 to 30, we multiply by 10. So, we multiply both the numerator and the denominator of 2/3 by 10:

(2/3) x (10/10) = 20/30

2. Convert 1/2 to an equivalent fraction with a denominator of 30:

To get from 2 to 30, we multiply by 15. So, we multiply both the numerator and the denominator of 1/2 by 15:

(1/2) x (15/15) = 15/30

3. Convert 8/15 to an equivalent fraction with a denominator of 30:

To get from 15 to 30, we multiply by 2. So, we multiply both the numerator and the denominator of 8/15 by 2:

(8/15) x (2/2) = 16/30

Now we have our original problem rewritten with a common denominator:

20/30 - 15/30 + 16/30

4. Perform the addition and subtraction:

Now that the fractions have the same denominator, we can simply add and subtract the numerators:

(20 - 15 + 16) / 30 = 21/30

Great! We've arrived at an answer. But we're not quite done yet. The final step is to simplify the fraction.

5. Simplify the fraction:

Simplifying a fraction means reducing it to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

In our case, we need to find the GCD of 21 and 30. Let's list the factors of each number:

  • Factors of 21: 1, 3, 7, 21
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

The greatest common factor of 21 and 30 is 3. So, we divide both the numerator and the denominator by 3:

(21/3) / (30/3) = 7/10

And there we have it! The fraction 21/30 simplified to its simplest form is 7/10. So, the final answer to our problem 2/3 - 1/2 + 8/15 is 7/10.

Tips and Tricks for Mastering Fraction Arithmetic

Fractions might seem daunting at first, but with consistent practice and the right approach, you can become a fraction master. Here are a few tips and tricks to help you along the way:

  • Practice regularly: Like any mathematical skill, proficiency in fractions comes with practice. The more you work with fractions, the more comfortable you'll become. Try solving different types of problems, from simple addition and subtraction to more complex multiplication and division.
  • Visualize fractions: Sometimes, it helps to visualize fractions. Think of them as parts of a whole, like slices of a pie or sections of a rectangle. This can make the concepts of adding, subtracting, and comparing fractions more intuitive.
  • Master the LCM and GCD: Understanding how to find the Least Common Multiple (LCM) and Greatest Common Divisor (GCD) is crucial for working with fractions. These concepts are fundamental to finding common denominators and simplifying fractions. Practice finding the LCM and GCD of different sets of numbers to strengthen your skills.
  • Break down complex problems: If you encounter a complex fraction problem, break it down into smaller, more manageable steps. This can make the problem less overwhelming and easier to solve. For example, if you have a problem with multiple operations, tackle them one at a time, following the order of operations (PEMDAS/BODMAS).
  • Check your work: Always take the time to check your work, especially when dealing with fractions. A small mistake can lead to a wrong answer. Double-check your calculations, make sure you've simplified your fractions correctly, and consider whether your answer makes sense in the context of the problem.

Conclusion: You've Got This!

So, there you have it! We've successfully navigated the world of fraction arithmetic and solved the problem 2/3 - 1/2 + 8/15, arriving at the simplified answer of 7/10. Remember, the key to mastering fractions is understanding the underlying concepts and practicing consistently. Don't be afraid to make mistakes – they're a natural part of the learning process. With each problem you solve, you'll build confidence and strengthen your skills. Keep practicing, and you'll be a fraction whiz in no time!