Fraction Calculations Step By Step Solutions And Guide

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Hey guys! Today, we're diving deep into the fascinating world of fraction calculations. Fractions can seem a bit intimidating at first, but trust me, with a little practice, you'll be solving them like a pro in no time! We'll break down each problem step-by-step, making sure you understand the why behind the how. So, grab your pencils, and let's get started!

a) (1/2) × (6 1/5 - 2/5)

Let's start with our first problem, which involves multiplying a fraction by the result of a subtraction within parentheses. This kind of problem tests your understanding of the order of operations (remember PEMDAS/BODMAS?) and your ability to work with mixed numbers and fractions. First, we need to rewrite the mixed number and calculate the numbers inside the parentheses. These initial calculations are key to simplifying the problem and making it easier to solve. Once we've simplified the expression inside the parentheses, the next step is to perform the multiplication. Multiplying fractions involves multiplying the numerators (the top numbers) and the denominators (the bottom numbers). After multiplying, we might need to simplify the resulting fraction. Simplifying fractions means reducing them to their lowest terms, which makes the answer cleaner and easier to understand. This often involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. Understanding these steps thoroughly is crucial not just for this problem, but for tackling any fraction calculation that comes your way. Mastering these skills ensures you have a solid foundation for more complex math problems later on.

So, let's break it down:

  1. Convert the mixed number to an improper fraction: 6 1/5 = (6 × 5 + 1) / 5 = 31/5
  2. Solve the subtraction inside the parentheses: (31/5 - 2/5) = 29/5
  3. Multiply: (1/2) × (29/5) = 29/10
  4. Convert back to a mixed number (optional): 29/10 = 2 9/10

b) (3 1/2 - 2 1/3) × 1 4/5

Next up, we have a problem that combines subtraction of mixed numbers with multiplication. This one's a bit more challenging, but don't worry, we'll conquer it together! The key here is to again remember the order of operations and to handle those mixed numbers with care. First things first, we need to convert those mixed numbers into improper fractions. This makes the subtraction and multiplication steps much easier to manage. Improper fractions allow us to perform arithmetic operations more directly without having to worry about the whole number part. Once we've converted the mixed numbers, we can tackle the subtraction within the parentheses. To subtract fractions, they need to have a common denominator. Finding the least common multiple (LCM) of the denominators is essential for efficient subtraction. After subtracting, we move on to the multiplication. As before, we multiply the numerators and the denominators. Finally, we simplify the resulting fraction, if possible, and convert it back to a mixed number for a more reader-friendly answer. By breaking down the problem into these manageable steps, we can avoid feeling overwhelmed and ensure accuracy in our calculations. Remember, practice makes perfect, and each problem you solve builds your confidence and skill.

Let's solve this step-by-step:

  1. Convert mixed numbers to improper fractions:
    • 3 1/2 = (3 × 2 + 1) / 2 = 7/2
    • 2 1/3 = (2 × 3 + 1) / 3 = 7/3
    • 1 4/5 = (1 × 5 + 4) / 5 = 9/5
  2. Solve the subtraction inside the parentheses (find a common denominator):
    • (7/2 - 7/3) = (21/6 - 14/6) = 7/6
  3. Multiply: (7/6) × (9/5) = 63/30
  4. Simplify and convert to a mixed number: 63/30 = 21/10 = 2 1/10

c) (2/9) × (9/10) + 2/3

Alright, let's tackle this problem, which involves both multiplication and addition of fractions. Remember, multiplication comes before addition in the order of operations, so that's where we'll start. When we multiply fractions, we simply multiply the numerators together and the denominators together. This step is straightforward, but it's important to be accurate to avoid errors later on. Once we've performed the multiplication, we move on to the addition. To add fractions, they need to have a common denominator. Finding the least common multiple (LCM) of the denominators is crucial for this step. The LCM allows us to rewrite the fractions with the same denominator, making addition possible. After finding the common denominator, we add the numerators while keeping the denominator the same. Finally, as always, we simplify the resulting fraction to its lowest terms. This ensures our answer is in the most concise and understandable form. Mastering this combination of multiplication and addition is a fundamental skill in fraction arithmetic, and it sets the stage for more complex calculations in the future.

Here's how we solve it:

  1. Multiply: (2/9) × (9/10) = 18/90
  2. Simplify the result of multiplication: 18/90 = 1/5
  3. Add (find a common denominator): (1/5 + 2/3) = (3/15 + 10/15) = 13/15

d) 3/5 + (2/5 - 3/10)

Let's dive into this problem, which combines addition and subtraction of fractions. The key here is to remember our friend PEMDAS/BODMAS and tackle the parentheses first. This means we'll start by subtracting the fractions inside the parentheses. To subtract fractions, they need a common denominator, so that's our first order of business. We'll find the least common multiple (LCM) of the denominators and rewrite the fractions accordingly. Once we've subtracted the fractions inside the parentheses, we're left with a simple addition problem. Again, we'll need a common denominator to add the fractions. After finding the common denominator, we add the numerators while keeping the denominator the same. Finally, we simplify the resulting fraction to its lowest terms. This step is important for presenting our answer in the clearest and most concise way. By following these steps carefully, we can confidently solve problems involving both addition and subtraction of fractions.

Let's break it down step-by-step:

  1. Solve the subtraction inside the parentheses (find a common denominator):
    • (2/5 - 3/10) = (4/10 - 3/10) = 1/10
  2. Add (find a common denominator): (3/5 + 1/10) = (6/10 + 1/10) = 7/10

e) 5/6 + 2 1/3 + 2 × 1/4

This problem is a fantastic mix of addition, multiplication, and a mixed number, giving us a great opportunity to practice multiple skills. We'll be leaning on the order of operations once again, so multiplication takes precedence. We'll tackle that multiplication first, and then we'll move on to the addition. Before we add, though, we need to address the mixed number. Converting it to an improper fraction will make the addition process much smoother. Once all our terms are in fraction form, we'll need to find a common denominator to add them together. This is a crucial step, ensuring that we're adding comparable parts. After we've added the fractions, we'll simplify the result, if possible, and convert it back to a mixed number for a clear and easy-to-understand final answer. This problem is a wonderful example of how different fraction operations can come together, and mastering it will boost your confidence in tackling more complex problems.

Here's the breakdown:

  1. Multiply: 2 × 1/4 = 2/4 = 1/2
  2. Convert the mixed number to an improper fraction: 2 1/3 = (2 × 3 + 1) / 3 = 7/3
  3. Add (find a common denominator): (5/6 + 7/3 + 1/2) = (5/6 + 14/6 + 3/6) = 22/6
  4. Simplify and convert to a mixed number: 22/6 = 11/3 = 3 2/3

f) 3/30 + 5/6 + 2/3 × 3/5

Okay, let's break down this problem, which involves a combination of addition and multiplication of fractions. As always, we need to keep the order of operations in mind, which means multiplication comes first. So, we'll start by multiplying the fractions 2/3 and 3/5. After we've tackled the multiplication, we're left with an addition problem involving three fractions. To add these fractions, we need to find a common denominator. This is a crucial step, as it allows us to combine the fractions correctly. Once we have a common denominator, we add the numerators while keeping the denominator the same. Finally, we simplify the resulting fraction to its lowest terms. This not only makes our answer cleaner but also demonstrates a strong understanding of fraction arithmetic. Problems like this one are great for reinforcing the importance of following the order of operations and mastering the techniques for adding and multiplying fractions.

Here's how we solve it step-by-step:

  1. Multiply: (2/3) × (3/5) = 6/15
  2. Simplify the result of multiplication: 6/15 = 2/5
  3. Add (find a common denominator): (3/30 + 5/6 + 2/5) = (3/30 + 25/30 + 12/30) = 40/30
  4. Simplify: 40/30 = 4/3 = 1 1/3

g) 5/6 + 2 1/3 + 2

Let's jump into this problem, which combines addition of a fraction, a mixed number, and a whole number. To make things easier, our first step is to convert the mixed number into an improper fraction. This puts all our numbers in a similar format, making the addition process smoother. We can also think of the whole number, 2, as a fraction with a denominator of 1 (2/1). Now we have three terms to add, and to do that, we need a common denominator. Finding the least common multiple (LCM) of the denominators is key to making this addition straightforward. Once we have a common denominator, we can add the numerators while keeping the denominator the same. Finally, we simplify the resulting fraction, if possible, and convert it back to a mixed number to present our answer in a clear and understandable way. This problem is a good reminder that fractions, mixed numbers, and whole numbers can all play together nicely when we use the right techniques.

Here's how we tackle it:

  1. Convert the mixed number to an improper fraction: 2 1/3 = (2 × 3 + 1) / 3 = 7/3
  2. Rewrite the whole number as a fraction: 2 = 2/1
  3. Add (find a common denominator): (5/6 + 7/3 + 2/1) = (5/6 + 14/6 + 12/6) = 31/6
  4. Convert to a mixed number: 31/6 = 5 1/6

h) 3 × 2/5 - 2 1/3 + 6

Lastly, let's dissect this problem, which brings together multiplication, subtraction, and addition, along with a mixed number. It's a real test of our understanding of the order of operations! Following PEMDAS/BODMAS, we'll start with the multiplication. After that, we'll need to convert the mixed number into an improper fraction to make the subtraction easier. Once we have all our terms in fraction or whole number form, we can proceed with the subtraction and addition from left to right. Remember, to subtract or add fractions, we need a common denominator. After performing the operations, we simplify the resulting fraction, if possible, and convert it to a mixed number if it's greater than one. This problem nicely demonstrates how all the different fraction operations can come together in a single calculation, and mastering it will give you a real sense of accomplishment!

Let's break it down step by step:

  1. Multiply: 3 × 2/5 = 6/5
  2. Convert the mixed number to an improper fraction: 2 1/3 = (2 × 3 + 1) / 3 = 7/3
  3. Subtract and add (find a common denominator):
    • (6/5 - 7/3 + 6) = (18/15 - 35/15 + 90/15) = 73/15
  4. Convert to a mixed number: 73/15 = 4 13/15

Key Takeaways

  • Order of Operations: Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
  • Mixed Numbers: Convert them to improper fractions for easier calculations.
  • Common Denominators: Essential for adding and subtracting fractions.
  • Simplifying: Always reduce your final answer to its simplest form.

I hope this comprehensive guide has helped you better understand fraction calculations. Remember, practice is key, so keep working on these problems, and you'll become a fraction master in no time! You got this!