Calculating 6/8 + 5/7: A Step-by-Step Guide
Hey guys! Today, we're diving into a basic math problem that many students often encounter: calculating the sum of two fractions, specifically 6/8 + 5/7. Don't worry, it's not as intimidating as it might seem. We'll break it down step by step so you can tackle similar problems with confidence. So, let's get started!
Understanding the Basics
Before we jump into the calculation, let's quickly recap what fractions are and how they work. A fraction represents a part of a whole. It consists of two numbers: the numerator (the number on top) and the denominator (the number on the bottom). The numerator tells you how many parts you have, and the denominator tells you how many parts the whole is divided into. In our problem, 6/8 means we have 6 parts out of a whole that's divided into 8 parts, and 5/7 means we have 5 parts out of a whole that's divided into 7 parts.
To add fractions, they need to have the same denominator, which is called a common denominator. This is because you can only add or subtract quantities that are measured in the same units. Think of it like trying to add apples and oranges – you need to convert them to a common unit (like "fruit") before you can add them together. Similarly, we need to find a common denominator for 8 and 7 before we can add 6/8 and 5/7.
Finding the Least Common Denominator (LCD)
The least common denominator (LCD) is the smallest number that both denominators can divide into evenly. In other words, it's the smallest common multiple of the denominators. To find the LCD of 8 and 7, we can list the multiples of each number until we find a common multiple:
- Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, ...
- Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, ...
As you can see, the smallest common multiple of 8 and 7 is 56. Therefore, the LCD is 56. Alternatively, since 8 and 7 don't share any common factors other than 1, you can simply multiply them together to find the LCD: 8 * 7 = 56. Understanding and finding the Least Common Denominator is very essential to solve this question. Without knowing the exact steps, the question would be hard to solve and would lead to mistakes. So make sure you understand the steps!
Converting the Fractions to Equivalent Fractions
Now that we have the LCD, we need to convert both fractions to equivalent fractions with a denominator of 56. An equivalent fraction is a fraction that has the same value as another fraction, but with a different numerator and denominator. To convert a fraction to an equivalent fraction, we multiply both the numerator and the denominator by the same number. For 6/8, we need to find a number that we can multiply 8 by to get 56. Since 8 * 7 = 56, we multiply both the numerator and the denominator of 6/8 by 7:
6/8 = (6 * 7) / (8 * 7) = 42/56
Similarly, for 5/7, we need to find a number that we can multiply 7 by to get 56. Since 7 * 8 = 56, we multiply both the numerator and the denominator of 5/7 by 8:
5/7 = (5 * 8) / (7 * 8) = 40/56
So, now we have two equivalent fractions: 42/56 and 40/56. These fractions have the same value as the original fractions, but they have a common denominator, which allows us to add them together.
Adding the Fractions
Now that we have two fractions with a common denominator, we can add them together. To add fractions with a common denominator, we simply add the numerators and keep the denominator the same:
42/56 + 40/56 = (42 + 40) / 56 = 82/56
So, the sum of 6/8 and 5/7 is 82/56. The concept of equivalent fractions is very useful and important. In reality, you will use this concept to find other fractions and not only to solve questions. In the future, you might even use it in real life when you are baking or cooking something!
Simplifying the Fraction (if possible)
The final step is to simplify the fraction, if possible. Simplifying a fraction means reducing it to its lowest terms by dividing both the numerator and the denominator by their greatest common factor (GCF). The GCF is the largest number that divides both the numerator and the denominator evenly. In our case, the GCF of 82 and 56 is 2. So, we divide both the numerator and the denominator by 2:
82/56 = (82 ÷ 2) / (56 ÷ 2) = 41/28
So, the simplified fraction is 41/28. Now, since the numerator (41) is greater than the denominator (28), we have an improper fraction. We can convert this to a mixed number. To do this, we divide 41 by 28:
41 ÷ 28 = 1 with a remainder of 13
This means that 41/28 is equal to 1 whole and 13/28. So, the mixed number is 1 13/28.
Final Answer
Therefore, 6/8 + 5/7 = 41/28 or 1 13/28. And that's it! You've successfully calculated the sum of two fractions. Remember to always find a common denominator, convert the fractions to equivalent fractions, add the numerators, and simplify the fraction if possible. Keep practicing, and you'll become a pro at adding fractions in no time! Good luck, and happy calculating!
Remember, practice makes perfect. The more you work with fractions, the easier it will become. Don't be afraid to make mistakes – they are a part of the learning process. Just keep trying, and you'll eventually master it. And hey, if you ever get stuck, there are plenty of resources available online and in textbooks to help you out. Keep up the great work, and never stop learning! It can be frustrating at times, but it's important not to give up. Breaking down the problem into smaller, more manageable steps can make it less overwhelming. Visual aids, like drawing diagrams or using fraction bars, can also be helpful. The best way to improve your skills is to practice regularly. Try working through different examples and gradually increasing the difficulty level. You can also find practice problems online or in textbooks. And of course, don't hesitate to ask for help from your teacher, classmates, or a tutor if you're struggling. Remember, everyone learns at their own pace, so be patient with yourself and celebrate your progress along the way.
