How To Solve 1/5 + 5/6 A Step-by-Step Guide

Hey guys! Today, we're diving into a super common math problem: adding fractions. Specifically, we're going to tackle the question, "What is the result of the operation $\frac{1}{5} + \frac{5}{6}$?" and explore the options provided. Adding fractions might seem tricky at first, but don't worry, I'm here to break it down in a way that's easy to understand. We'll walk through the process together, step by step, so you can confidently solve similar problems in the future. So, let's put on our math hats and get started!

Understanding the Challenge: Why Can't We Just Add Across?

Before we jump into solving the problem, it's crucial to understand why we can't simply add the numerators (the top numbers) and the denominators (the bottom numbers) straight away. This is a common mistake people make, so let's clear it up right now. Imagine you're trying to add a slice of pizza that's one-fifth of the whole pie to another slice that's five-sixths of a different (or even the same) pie. It's like trying to add apples and oranges – they're different units. To add fractions correctly, they need to have the same denominator, representing the same size of "slice." Think of it like this: if both fractions are "out of 30," then we're comparing the same sized pieces, and we can easily see how many total pieces we have.

This shared denominator is called the least common denominator (LCD). Finding the LCD is the key to adding fractions, and it's not as scary as it sounds! We need to find the smallest number that both 5 and 6 divide into evenly. There are different ways to find the LCD, but we'll cover a straightforward method in the next section. Mastering the concept of the LCD is fundamental not only for this specific problem but for countless other fraction-related calculations you'll encounter in math. It's a foundational skill that will serve you well in algebra, geometry, and even everyday life situations involving proportions and ratios. So, pay close attention, and let's get this LCD thing figured out!

Finding the Common Ground: The Least Common Denominator (LCD)

Okay, so we know we need a common denominator, but how do we find it? The easiest way to find the least common denominator (LCD) is to list out the multiples of each denominator until we find one they share. Let's do that for 5 and 6:

• Multiples of 5: 5, 10, 15, 20, 25, 30, 35...
• Multiples of 6: 6, 12, 18, 24, 30, 36...

Bingo! We see that 30 is the smallest number that appears in both lists. This means 30 is our LCD. Another way to find the LCD, especially for larger numbers, is to use the prime factorization method. You break down each number into its prime factors and then take the highest power of each prime factor that appears in either number. While this method is very reliable, for simple cases like 5 and 6, listing out the multiples is often quicker. Now that we've found our LCD, the real fun begins! We need to rewrite our fractions so they both have a denominator of 30. This involves multiplying both the numerator and the denominator of each fraction by a specific number. Remember, whatever you do to the bottom, you have to do to the top! This is crucial to maintain the fraction's value. We're essentially creating equivalent fractions, fractions that look different but represent the same amount. Think of it like slicing a pizza: whether you cut it into 5 slices or 30 slices, the whole pizza is still the same amount of pizza. We're just changing how we represent it. So, let's get to work on rewriting those fractions!

Transforming the Fractions: Making Them Compatible

Now that we know our LCD is 30, let's transform our fractions, $\frac{1}{5}$ and $\frac{5}{6}$, so they both have a denominator of 30. To do this, we need to figure out what to multiply the denominator of each fraction by to get 30. For $\frac{1}{5}$, we need to multiply the denominator, 5, by 6 because 5 * 6 = 30. Remember the rule: whatever we do to the bottom, we must also do to the top. So, we multiply the numerator, 1, by 6 as well. This gives us:

$\frac{1}{5} * \frac{6}{6} = \frac{6}{30}$

See? We've created an equivalent fraction! $\frac{1}{5}$ is the same as $\frac{6}{30}$. Now, let's do the same for $\frac{5}{6}$. We need to multiply the denominator, 6, by 5 because 6 * 5 = 30. Again, we must multiply the numerator, 5, by the same number, 5. This gives us:

$\frac{5}{6} * \frac{5}{5} = \frac{25}{30}$

Awesome! We've successfully transformed both fractions to have the same denominator. $\frac{1}{5}$ has become $\frac{6}{30}$, and $\frac{5}{6}$ has become $\frac{25}{30}$. Now that they have a common denominator, we can finally add them together. This is where the magic happens! We've laid the groundwork, and the rest is smooth sailing. So, let's move on to the next step and actually add those fractions.

The Grand Finale: Adding and Simplifying (If Necessary)

Alright, we've done the hard work of finding the LCD and transforming our fractions. Now comes the easy (and satisfying) part: adding them! We have $\frac{6}{30}$ and $\frac{25}{30}$. When fractions have the same denominator, we simply add the numerators and keep the denominator the same. So, we have:

$\frac{6}{30} + \frac{25}{30} = \frac{6 + 25}{30} = \frac{31}{30}$

And there you have it! The sum of $\frac{1}{5}$ and $\frac{5}{6}$ is $\frac{31}{30}$. But hold on a second! It's always a good practice to check if our answer can be simplified. In this case, $\frac{31}{30}$ is an improper fraction because the numerator (31) is greater than the denominator (30). This means we can convert it to a mixed number, which might be a more intuitive way to understand the quantity. To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator stays the same. So, 31 divided by 30 is 1 with a remainder of 1. This means $\frac{31}{30}$ is equal to 1 $\frac{1}{30}$. While $\frac{31}{30}$ is a perfectly valid answer, expressing it as a mixed number sometimes helps us visualize the amount better. In our case, it tells us that the sum is slightly more than 1. Now, let's go back to the original question and see which of the options matches our answer.

Choosing the Right Path: Identifying the Correct Option

We've successfully calculated that $\frac{1}{5} + \frac{5}{6} = \frac{31}{30}$. Now, let's revisit the options provided in the original question and see which one matches our result:

a) $\frac{13}{6}$ b) $\frac{31}{30}$ c) $\frac{11}{30}$ d) $\frac{29}{30}$

It's clear as day! Option b) $\frac{31}{30}$ is the correct answer. We did it! We took on the challenge of adding fractions with different denominators, found the LCD, transformed the fractions, added them, and even considered simplifying our answer. This is a fantastic example of how breaking down a problem into smaller, manageable steps can make even seemingly complex tasks much easier. Remember, math is all about understanding the underlying principles and applying them systematically. By understanding the concept of the LCD and how to create equivalent fractions, you've equipped yourself with a valuable tool for solving a wide range of fraction problems. So, give yourself a pat on the back, and let's keep exploring the fascinating world of mathematics!

Key Takeaways: Mastering Fraction Addition

Before we wrap things up, let's quickly recap the key takeaways from this problem. Mastering these concepts will not only help you solve similar problems but also build a solid foundation for more advanced math topics. First and foremost, remember that you cannot directly add fractions unless they have a common denominator. This is the golden rule of fraction addition! Secondly, the least common denominator (LCD) is the smallest number that both denominators divide into evenly. Finding the LCD is the crucial first step in adding fractions with different denominators. Thirdly, to transform fractions to have the LCD, you need to multiply both the numerator and the denominator by the same number. This creates an equivalent fraction, which looks different but represents the same value. Fourthly, once the fractions have a common denominator, you can simply add the numerators and keep the denominator the same. Finally, always check if your answer can be simplified, either by reducing the fraction to its lowest terms or by converting an improper fraction to a mixed number. By keeping these key takeaways in mind, you'll be well-equipped to tackle any fraction addition problem that comes your way. So, keep practicing, keep exploring, and remember that math can be fun and rewarding!

What is the sum of $\frac{1}{5}$ and $\frac{5}{6}$? Choose from the following options: a) $\frac{13}{6}$ b) $\frac{31}{30}$ c) $\frac{11}{30}$ d) $\frac{29}{30}$

Solving Fractions: How to Calculate 1/5 + 5/6 with Ease