How To Solve 5/7 + 4/5? A Step-by-Step Guide

by Henrik Larsen 45 views

Hey guys! Let's dive into the fascinating world of fraction addition, specifically tackling the problem 5/7 + 4/5. Adding fractions might seem daunting at first, but trust me, with the right approach, it becomes a piece of cake. This guide will walk you through each step, ensuring you not only understand the process but also master it. We'll break down the problem, discuss the underlying principles, and explore why each step is crucial. So, buckle up and get ready to conquer fraction addition!

Understanding the Basics of Fraction Addition

Before we jump into the specifics of adding 5/7 and 4/5, it's essential to grasp the fundamental principles of fraction addition. Remember, fractions represent parts of a whole. The top number (numerator) indicates how many parts we have, and the bottom number (denominator) indicates the total number of parts that make up the whole. When adding fractions, we're essentially combining these parts. However, we can only directly add fractions if they have the same denominator. This is because we need to be adding parts of the same "size." Think of it like trying to add apples and oranges – you can't simply add the numbers because they're different units. You need a common unit, like "fruit," to make the addition meaningful. Similarly, with fractions, we need a common denominator to add them accurately.

So, why is a common denominator so important? Imagine you have a pizza cut into 7 slices, and you take 5 slices (5/7 of the pizza). Then, you have another pizza cut into 5 slices, and you take 4 slices (4/5 of the pizza). To figure out the total amount of pizza you have, you can't just add the numerators (5 + 4) because the slices are different sizes. The slices from the first pizza are smaller than the slices from the second pizza. To accurately add them, you need to find a way to express both fractions with the same size slices – that's where the common denominator comes in. Finding the common denominator allows us to express both fractions in terms of the same "unit," making addition possible.

The Key Concept: Common Denominator

To add fractions, the most important concept to remember is the common denominator. This is the shared bottom number that allows us to accurately combine the fractions. Think of it as the common language that both fractions need to speak before they can be added together. Without it, we're trying to add unlike terms, which simply doesn't work. The common denominator ensures that we're adding equal-sized pieces, giving us a meaningful result. This fundamental principle underpins the entire process of fraction addition, and mastering it is crucial for success.

Finding the Least Common Denominator (LCD)

Now that we understand why a common denominator is crucial, let's talk about how to find it. For the problem 5/7 + 4/5, we need to find the Least Common Denominator (LCD). The LCD is the smallest multiple that both denominators (7 and 5) share. There are a couple of ways to find the LCD:

  1. Listing Multiples: We can list the multiples of each denominator until we find a common one.

    • Multiples of 7: 7, 14, 21, 28, 35, 42...
    • Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40... The first common multiple we find is 35, so the LCD is 35.

  2. Prime Factorization: We can find the prime factorization of each denominator and then multiply the highest power of each prime factor.

    • Prime factorization of 7: 7 (7 is a prime number)
    • Prime factorization of 5: 5 (5 is a prime number) Since 7 and 5 are both prime, their LCD is simply their product: 7 * 5 = 35.

In this case, both methods lead us to the same conclusion: the LCD of 7 and 5 is 35. This means we need to rewrite both fractions so that they have a denominator of 35. This step is absolutely crucial, guys! If you don't get the LCD right, the rest of the calculation will be incorrect. Take your time, use whichever method you prefer, and double-check your work to ensure accuracy. The LCD is the foundation upon which the rest of the solution is built.

Why the Least Common Denominator?

You might be wondering, why the least common denominator? Why not just use any common denominator? While using any common denominator will eventually lead to the correct answer, using the LCD makes the process simpler and keeps the numbers smaller. Smaller numbers are easier to work with, reducing the chances of making mistakes. Additionally, using the LCD ensures that the final answer is in its simplest form, requiring fewer steps for simplification later on. So, while technically any common denominator will work, the LCD is the most efficient and practical choice.

Converting Fractions to Equivalent Fractions with the LCD

Now that we've found the LCD (which is 35 for 5/7 + 4/5), we need to convert each fraction into an equivalent fraction with the denominator of 35. This means we need to find new numerators for each fraction while maintaining their value. Remember, we can't just change the denominator without also changing the numerator; otherwise, we're changing the fraction itself!

To convert 5/7 to an equivalent fraction with a denominator of 35, we need to figure out what number to multiply the denominator (7) by to get 35. In this case, 7 * 5 = 35. So, we multiply both the numerator and the denominator of 5/7 by 5:

(5 * 5) / (7 * 5) = 25/35

Similarly, to convert 4/5 to an equivalent fraction with a denominator of 35, we need to figure out what number to multiply the denominator (5) by to get 35. In this case, 5 * 7 = 35. So, we multiply both the numerator and the denominator of 4/5 by 7:

(4 * 7) / (5 * 7) = 28/35

Now we have two equivalent fractions: 25/35 and 28/35. Notice that 25/35 is equivalent to 5/7, and 28/35 is equivalent to 4/5. We haven't changed the value of the fractions; we've simply expressed them with a common denominator. This step is crucial for accurate addition. By ensuring both fractions have the same denominator, we're now adding equal-sized parts, making the addition process meaningful and straightforward.

Maintaining Equivalence: The Golden Rule of Fractions

The key to converting fractions correctly is to remember the golden rule of fractions: Whatever you do to the denominator, you must do to the numerator, and vice versa. This ensures that you're not changing the value of the fraction, only its representation. Multiplying both the numerator and denominator by the same number is essentially multiplying the fraction by 1 (e.g., 5/5 or 7/7), which doesn't change its value. This principle is fundamental to working with fractions and is essential for accurate calculations.

Adding the Equivalent Fractions

With our fractions now sharing a common denominator, we can finally add them! We have 25/35 + 28/35. The rule for adding fractions with the same denominator is simple: add the numerators and keep the denominator the same.

So, 25/35 + 28/35 = (25 + 28) / 35 = 53/35

That's it! We've added the fractions. Our initial calculation gives us 53/35. This fraction represents the sum of 5/7 and 4/5. However, there's one more step we might need to take: simplifying the fraction.

Why Keep the Denominator the Same?

It's important to understand why we keep the denominator the same when adding fractions. Remember, the denominator represents the total number of parts that make up the whole, and the numerator represents how many of those parts we have. When we add fractions with the same denominator, we're essentially combining the number of parts we have, but the size of those parts (represented by the denominator) remains the same. Think back to our pizza analogy: if you have 2 slices of a pizza cut into 8 slices each (2/8) and someone gives you 3 more slices from the same pizza (3/8), you now have 5 slices (5/8). The pizza is still cut into 8 slices, so the denominator remains 8. We only added the numerators because we were adding slices of the same size.

Simplifying the Resulting Fraction

Our result, 53/35, is an improper fraction because the numerator (53) is greater than the denominator (35). While this answer is technically correct, it's often preferable to express it as a mixed number or simplify it if possible. In this case, we'll convert the improper fraction to a mixed number.

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, 53 ÷ 35 = 1 with a remainder of 18.

This means 53/35 is equal to 1 and 18/35. We write this as a mixed number: 1 18/35.

Now, we need to check if the fractional part (18/35) can be simplified further. To do this, we look for common factors between the numerator (18) and the denominator (35). The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 35 are 1, 5, 7, and 35. The only common factor is 1, which means 18/35 is already in its simplest form.

Therefore, the simplified result of 5/7 + 4/5 is 1 18/35. We've successfully added the fractions and expressed the result in its simplest form!

Why Simplify?

Simplifying fractions is crucial for several reasons. First, it presents the answer in its most understandable form. A mixed number often provides a clearer sense of the quantity than an improper fraction. Second, simplified fractions are easier to compare and work with in further calculations. Imagine trying to compare 53/35 to another fraction – it's much easier to compare 1 18/35. Finally, simplifying fractions is often expected in mathematical contexts, making it a necessary skill to master.

Conclusion: Mastering Fraction Addition

Congratulations, guys! You've successfully navigated the process of adding fractions, specifically 5/7 + 4/5. We've covered the importance of the common denominator, the steps for finding the LCD, converting fractions, adding numerators, and simplifying the result. Remember, practice makes perfect, so don't hesitate to tackle more fraction addition problems to solidify your understanding. With a solid grasp of these principles, you'll be able to confidently add fractions of all kinds. Keep practicing, and you'll become a fraction addition pro in no time!

Key Takeaways

  • Adding fractions requires a common denominator.
  • The Least Common Denominator (LCD) is the smallest multiple shared by the denominators.
  • Convert fractions to equivalent fractions with the LCD before adding.
  • Add the numerators and keep the denominator the same.
  • Simplify the resulting fraction to its simplest form.