Solving 7/5 + 2 1/3: A Step-by-Step Guide

by Dimemap Team 42 views

Hey guys! Ever get a math problem that looks like a jumble of fractions and mixed numbers? Well, let's tackle one together! We're going to break down how to solve 7/5 + 2 1/3 in a way that’s super easy to understand. Trust me, by the end of this, you’ll be a fraction-adding pro!

Understanding the Problem

Before we dive into the nitty-gritty, let's make sure we understand what the problem is asking. We have two numbers here: a fraction (7/5) and a mixed number (2 1/3). Our mission is to add these two together. The key to success here is knowing how to convert and manipulate these numbers so we can add them smoothly. So, what are fractions and mixed numbers anyway? Let's do a quick recap.

  • Fractions: These represent a part of a whole. The top number (numerator) tells us how many parts we have, and the bottom number (denominator) tells us how many parts make up the whole. For example, 7/5 means we have 7 parts, and it takes 5 parts to make a whole. Notice that 7/5 is an improper fraction because the numerator is larger than the denominator, meaning it's more than one whole.
  • Mixed Numbers: These are a combination of a whole number and a fraction, like our 2 1/3. It means we have 2 whole units plus an additional 1/3. Mixed numbers are super common in everyday life, like when you're measuring ingredients for a recipe or figuring out how much time you have left.

Now that we're all clear on what fractions and mixed numbers are, let’s get this show on the road and start solving our problem!

Step 1: Convert the Mixed Number to an Improper Fraction

The first thing we need to do is make our lives easier by converting that mixed number (2 1/3) into an improper fraction. Why? Because it's much simpler to add fractions when they're both in the same format. Here’s how we do it:

  1. Multiply the whole number by the denominator: In our case, we multiply 2 (the whole number) by 3 (the denominator). That's 2 * 3 = 6.
  2. Add the numerator to the result: We then add the numerator, which is 1, to our previous result. So, 6 + 1 = 7.
  3. Keep the same denominator: The denominator stays the same, which is 3.

So, 2 1/3 becomes 7/3. Awesome! We've turned a mixed number into an improper fraction. Now our problem looks like this: 7/5 + 7/3. We're one step closer to solving it!

Step 2: Find the Least Common Denominator (LCD)

Alright, we’ve got two fractions now: 7/5 and 7/3. But we can't just add them as they are because they have different denominators. It's like trying to add apples and oranges – you need a common unit! That's where the Least Common Denominator (LCD) comes in. The LCD is the smallest number that both denominators (5 and 3) can divide into evenly. Think of it as finding the smallest “common ground” for our fractions.

So, how do we find the LCD? There are a couple of ways to do it:

  • Listing Multiples: Write out the multiples of each denominator until you find a common one.
    • Multiples of 5: 5, 10, 15, 20...
    • Multiples of 3: 3, 6, 9, 12, 15...
    • Aha! We see that 15 is a common multiple.
  • Prime Factorization: Break down each denominator into its prime factors and then multiply the highest power of each prime factor.
    • 5 is a prime number, so its prime factorization is just 5.
    • 3 is also a prime number, so its prime factorization is 3.
    • Multiply the prime factors: 5 * 3 = 15.

Either way, we find that the LCD for 5 and 3 is 15. This means we need to rewrite both fractions so they have a denominator of 15. Get ready for the next step!

Step 3: Convert Fractions to Equivalent Fractions with the LCD

Now that we've found our LCD (which is 15), it's time to rewrite our fractions so they both have this denominator. We're not changing the value of the fractions, just how they look. We're creating equivalent fractions. Here’s how we do it for each fraction:

  • For 7/5:

    1. Divide the LCD by the original denominator: 15 Ă· 5 = 3
    2. Multiply both the numerator and denominator by the result: (7 * 3) / (5 * 3) = 21/15

  • For 7/3:

    1. Divide the LCD by the original denominator: 15 Ă· 3 = 5
    2. Multiply both the numerator and denominator by the result: (7 * 5) / (3 * 5) = 35/15

Boom! We've successfully converted our fractions. Now we have 21/15 and 35/15. See how they both have the same denominator? This is exactly what we wanted! Now we can finally add them together.

Step 4: Add the Fractions

This is the moment we've been waiting for! Now that our fractions have the same denominator, adding them is a piece of cake. All we need to do is add the numerators and keep the denominator the same.

So, we have 21/15 + 35/15. Let’s add those numerators:

21 + 35 = 56

The denominator stays the same, which is 15.

So, 21/15 + 35/15 = 56/15. We've added the fractions! But we're not quite done yet. Our answer is currently an improper fraction, and it’s good practice to simplify it.

Step 5: Simplify the Improper Fraction

Our answer is 56/15, which is an improper fraction (the numerator is bigger than the denominator). We need to convert this back into a mixed number to make it easier to understand. Here’s how:

  1. Divide the numerator by the denominator: 56 Ă· 15 = 3 with a remainder of 11.
  2. The quotient (3) becomes the whole number: We have 3 whole units.
  3. The remainder (11) becomes the numerator of the fraction: We have 11 parts left over.
  4. The denominator (15) stays the same: Our fraction part is 11/15.

So, 56/15 is equal to 3 11/15. And guess what? We’ve got our final answer!

Final Answer

The result of 7/5 + 2 1/3 is 3 11/15.

Wow, we did it! We took a problem with fractions and mixed numbers and broke it down step by step. Remember, the key is to convert to improper fractions, find the LCD, create equivalent fractions, add the numerators, and simplify. You've got this! Keep practicing, and you'll be a fraction-solving whiz in no time! 🚀