Simplifying Fractions: A Step-by-Step Guide To 2/5 + 3/4

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Hey guys! Let's dive into simplifying fractions, specifically the addition problem 2/5 + 3/4. This is a fundamental concept in mathematics, and mastering it will help you tackle more complex problems down the road. So, grab your thinking caps, and let's get started!

Understanding the Basics of Fraction Addition

Before we jump into the solution, let’s make sure we understand the core concept behind fraction addition. The key idea here is that you can only add fractions directly if they have the same denominator. Think of the denominator as the 'unit' we're counting. If you're trying to add apples and oranges, you first need to express them in a common unit, like 'fruit.' Similarly, with fractions, we need a common denominator.

Why a Common Denominator Matters

Imagine you have a pizza cut into 5 slices, and you take 2 slices (2/5 of the pizza). Then, another pizza is cut into 4 slices, and you take 3 slices (3/4 of the pizza). You can't directly add 2 and 3 because they represent different-sized slices. To find the total amount of pizza you have, you need to express both fractions with a common denominator, meaning you need to divide both pizzas into slices of the same size.

Finding the Least Common Denominator (LCD)

The most efficient way to find a common denominator is to determine the Least Common Denominator (LCD). The LCD is the smallest number that is a multiple of both denominators. In our case, the denominators are 5 and 4. To find the LCD, you can list the multiples of each number:

  • Multiples of 5: 5, 10, 15, 20, 25...
  • Multiples of 4: 4, 8, 12, 16, 20, 24...

The smallest number that appears in both lists is 20. Therefore, the LCD of 5 and 4 is 20. This means we'll convert both fractions to have a denominator of 20.

Step-by-Step Solution: 2/5 + 3/4

Now that we understand the concept and have found the LCD, let's walk through the steps to solve 2/5 + 3/4.

Step 1: Convert Fractions to Equivalent Fractions with the LCD

To convert a fraction to an equivalent fraction with a different denominator, you multiply both the numerator and the denominator by the same number. This doesn't change the value of the fraction, only its representation.

  • Converting 2/5: To get a denominator of 20, we need to multiply 5 by 4. So, we multiply both the numerator and denominator of 2/5 by 4: (2 * 4) / (5 * 4) = 8/20
  • Converting 3/4: To get a denominator of 20, we need to multiply 4 by 5. So, we multiply both the numerator and denominator of 3/4 by 5: (3 * 5) / (4 * 5) = 15/20

Now we have two equivalent fractions: 8/20 and 15/20.

Step 2: Add the Fractions

With the fractions now having the same denominator, we can add them directly. To add fractions with the same denominator, you simply add the numerators and keep the denominator the same:

8/20 + 15/20 = (8 + 15) / 20 = 23/20

So, 2/5 + 3/4 = 23/20

Step 3: Simplify the Fraction (if possible)

Our result is 23/20. Now, let's see if we can simplify this fraction. A fraction can be simplified if the numerator and denominator have a common factor other than 1. In this case, 23 is a prime number (only divisible by 1 and itself), and it doesn't share any common factors with 20. Therefore, 23/20 is already in its simplest form. However, we can express it as a mixed number.

Converting an Improper Fraction to a Mixed Number

23/20 is an improper fraction because the numerator (23) is greater than the denominator (20). To convert it to a mixed number, we divide the numerator by the denominator:

23 ÷ 20 = 1 with a remainder of 3

This means 23/20 is equal to 1 whole and 3/20. So, we can write it as the mixed number 1 3/20.

The Answer and Why It Matters

Therefore, the simplified answer to 2/5 + 3/4 is 23/20 (or 1 3/20). Looking back at the options provided:

  • A. 5/9 (Incorrect)
  • B. 6/15 (Incorrect)
  • C. 23/20 (Correct)
  • D. 7/18 (Incorrect)

Option C is the correct answer.

Understanding how to add fractions is crucial for many areas of math, including algebra, geometry, and calculus. It's also a valuable skill in everyday life, whether you're measuring ingredients for a recipe, calculating proportions, or managing finances.

Common Mistakes to Avoid

Let's quickly touch on some common mistakes people make when adding fractions, so you can avoid them!

  • Adding numerators and denominators directly: This is a big no-no! You can't just add the top numbers and the bottom numbers separately unless the denominators are the same. Remember, you need a common denominator first.
  • Forgetting to multiply the numerator: When you multiply the denominator to get the LCD, you must also multiply the numerator by the same number. This is what creates the equivalent fraction.
  • Not simplifying the final answer: Always check if your answer can be simplified. It's good practice to express fractions in their simplest form.

Practice Makes Perfect

The best way to master fraction addition is to practice! Try solving different fraction addition problems. Start with simple ones and gradually work your way up to more complex problems. You can find plenty of practice problems online or in math textbooks.

Example Practice Problems:

  • 1/3 + 1/2
  • 3/8 + 1/4
  • 5/6 + 2/9
  • 7/10 + 1/5

Work through these problems, and remember the steps we discussed: find the LCD, convert to equivalent fractions, add the numerators, and simplify if possible.

Conclusion

Adding fractions might seem a bit tricky at first, but with a solid understanding of the concepts and a little practice, you'll become a pro in no time! Remember the importance of the common denominator, and always double-check your work. Keep practicing, and you'll be simplifying fractions like a math whiz. You got this! Keep up the great work, guys!