Mastering Fraction Calculations A Step-by-Step Guide

Hey guys! Ever feel like you're wrestling with fractions? Don't worry, you're not alone! Fractions can seem intimidating, but with a little practice and the right approach, you can conquer them like a math ninja. In this guide, we're going to break down two fraction calculation problems step-by-step, so you can build your confidence and skills. Let's dive in!

Unlocking Fraction Mastery with Step-by-Step Solutions

Problem (a) Decoding Complex Fraction Operations

Let's tackle our first calculation: (a) ⁸⁄₇ - 6 ¹/₂ + (1 ²/₃ - ⁷/₂) × ⁷/₆. This looks like a mouthful, right? But don't sweat it! We'll break it down into manageable steps using the order of operations (PEMDAS/BODMAS). Remember, this means we handle parentheses/brackets first, then exponents/orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

Step 1: Tackling the Parentheses

Our first mission is to simplify the expression inside the parentheses: **(1 ²/₃ - ⁷/₂) **. To do this, we need to convert the mixed number (1 ²/₃) into an improper fraction. Remember, an improper fraction has a numerator larger than or equal to the denominator. To convert, we multiply the whole number (1) by the denominator (3) and add the numerator (2), keeping the same denominator. So, 1 ²/₃ becomes ((1 * 3) + 2) / 3 = ⁵/₃. Now our expression looks like this: **(⁵/₃ - ⁷/₂) **.

Next, we need to subtract these fractions. But here's the catch: we can only add or subtract fractions if they have the same denominator. So, we need to find the least common multiple (LCM) of 3 and 2. The LCM is the smallest number that both 3 and 2 divide into evenly. In this case, the LCM is 6. Now, we need to convert both fractions to have a denominator of 6. To convert ⁵/₃, we multiply both the numerator and denominator by 2 (because 3 * 2 = 6). This gives us ¹⁰/₆. To convert ⁷/₂, we multiply both the numerator and denominator by 3 (because 2 * 3 = 6). This gives us ²¹/₆. Now we can rewrite our expression inside the parentheses as (¹⁰/₆ - ²¹/₆). Subtracting the numerators, we get -¹¹/₆. So, the simplified expression inside the parentheses is -¹¹/₆. Whew! That's the first hurdle cleared.

Step 2: Multiplication Time

Now our original problem looks like this: ⁸/₇ - 6 ¹/₂ + (-¹¹/₆) × ⁷/₆. Next up is the multiplication: (-¹¹/₆) × ⁷/₆. To multiply fractions, we simply multiply the numerators together and the denominators together. So, (-11 * 7) / (6 * 6) = -⁷⁷/₃₆. Remember that a negative times a positive is a negative. So now our problem is: ⁸/₇ - 6 ¹/₂ - ⁷⁷/₃₆.

Step 3: Mixed Number Conversion

Before we can add and subtract, let's convert the mixed number 6 ¹/₂ into an improper fraction. Using the same method as before, we get ((6 * 2) + 1) / 2 = ¹³/₂. Our problem now reads: ⁸/₇ - ¹³/₂ - ⁷⁷/₃₆.

Step 4: Finding the Common Denominator

To add and subtract these fractions, we need a common denominator. This time, we need to find the LCM of 7, 2, and 36. This might seem daunting, but let's break it down. The prime factors of 7 are just 7. The prime factors of 2 are 2. The prime factors of 36 are 2 x 2 x 3 x 3. To find the LCM, we take the highest power of each prime factor that appears in any of the numbers: 2² x 3² x 7 = 252. So, our common denominator is 252. This means we need to convert each fraction to have a denominator of 252.

Step 5: Fraction Conversion and Final Calculation

Let's convert each fraction: ⁸/₇ becomes (8 * 36) / (7 * 36) = ²⁸⁸/₂₅₂. ¹³/₂ becomes (13 * 126) / (2 * 126) = ¹⁶³⁸/₂₅₂. ⁷⁷/₃₆ becomes (77 * 7) / (36 * 7) = ⁵³⁹/₂₅₂. Now our problem is: ²⁸⁸/₂₅₂ - ¹⁶³⁸/₂₅₂ - ⁵³⁹/₂₅₂. Performing the subtraction, we get (288 - 1638 - 539) / 252 = -¹⁸⁸⁹/₂₅₂. This is an improper fraction, so let's convert it back to a mixed number. We divide 1889 by 252, which gives us 7 with a remainder of 125. So, our final answer is -7 ¹²⁵/₂₅₂. Boom! We conquered that complex fraction problem.

Problem (b) Navigating Multi-Term Fraction Expressions

Now, let's tackle the second calculation: **(b) 26 ¹/₂ - 3 ¾ - (2 ²/₃ + 1 ¹/₂) **. This one involves more mixed numbers and a set of parentheses. But we'll use the same strategies as before: order of operations, converting mixed numbers to improper fractions, finding common denominators, and simplifying.

Step 1: Conquering the Parentheses

First up, we need to simplify the expression inside the parentheses: **(2 ²/₃ + 1 ¹/₂) **. Let's convert the mixed numbers to improper fractions. 2 ²/₃ becomes ((2 * 3) + 2) / 3 = ⁸/₃. 1 ¹/₂ becomes ((1 * 2) + 1) / 2 = ³/₂. So, our expression inside the parentheses is now **(⁸/₃ + ³/₂) **. To add these fractions, we need a common denominator. The LCM of 3 and 2 is 6. Let's convert: ⁸/₃ becomes (8 * 2) / (3 * 2) = ¹⁶/₆. ³/₂ becomes (3 * 3) / (2 * 3) = ⁹/₆. Now we have (¹⁶/₆ + ⁹/₆). Adding the numerators, we get ²⁵/₆. So, the simplified expression inside the parentheses is ²⁵/₆. Great job!

Step 2: Mixed Number Conversions

Now our original problem looks like this: 26 ¹/₂ - 3 ¾ - ²⁵/₆. Let's convert the remaining mixed numbers to improper fractions. 26 ¹/₂ becomes ((26 * 2) + 1) / 2 = ⁵³/₂. 3 ¾ becomes ((3 * 4) + 3) / 4 = ¹⁵/₄. Our problem now reads: ⁵³/₂ - ¹⁵/₄ - ²⁵/₆.

Step 3: Finding the Common Denominator (Again!)

To subtract these fractions, we need a common denominator. We need to find the LCM of 2, 4, and 6. The prime factors of 2 are 2. The prime factors of 4 are 2 x 2. The prime factors of 6 are 2 x 3. The LCM is 2² x 3 = 12. So, our common denominator is 12. Time to convert those fractions!

Step 4: Fraction Conversion and the Final Showdown

Let's convert: ⁵³/₂ becomes (53 * 6) / (2 * 6) = ³¹⁸/₁₂. ¹⁵/₄ becomes (15 * 3) / (4 * 3) = ⁴⁵/₁₂. ²⁵/₆ becomes (25 * 2) / (6 * 2) = ⁵⁰/₁₂. Now our problem is: ³¹⁸/₁₂ - ⁴⁵/₁₂ - ⁵⁰/₁₂. Performing the subtraction, we get (318 - 45 - 50) / 12 = ²²³/₁₂. This is an improper fraction, so let's convert it back to a mixed number. We divide 223 by 12, which gives us 18 with a remainder of 7. So, our final answer is 18 ⁷/₁₂. Another fraction problem bites the dust!

The Fraction Calculation Toolkit Your Key Takeaways

Okay, guys, we've battled some pretty complex fraction problems. Here's what we've learned and what you should remember:

• Order of Operations is Your Friend: Always follow PEMDAS/BODMAS to ensure you're tackling operations in the correct order. This is super important for accuracy!
• Mixed Numbers Need a Makeover: Convert mixed numbers to improper fractions before performing addition, subtraction, multiplication, or division. It simplifies the process significantly.
• Common Denominators are Essential: You can only add or subtract fractions if they share a common denominator. Find the LCM to make your life easier. This is the golden rule of fraction addition and subtraction.
• Simplify, Simplify, Simplify: Always simplify your final answer, whether it's reducing an improper fraction to a mixed number or reducing the fraction to its lowest terms. It's like the final polish on a masterpiece!

Practice Makes Perfect Your Fraction Journey Continues

Fractions can be tricky, but with consistent practice, you'll become a fraction master in no time. Try working through similar problems on your own, and don't be afraid to make mistakes. Mistakes are learning opportunities! The more you practice, the more confident you'll become. So, keep those fractions coming, and keep sharpening your math skills!