Solving 2/5 × -3/5 - 1/14 - 1/13 × 3/5 A Step-by-Step Guide

by BRAINLY IN FTUNILA 60 views
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Hey there, math enthusiasts! Today, we're diving into a fascinating mathematical problem: 2/5 × -3/5 - 1/14 - 1/13 × 3/5. This expression might seem daunting at first glance, but don't worry, guys! We'll break it down step by step, unraveling its complexities and arriving at the solution with clarity and confidence. So, grab your thinking caps, and let's embark on this math expedition together!

Deciphering the Order of Operations: A Crucial First Step

Before we jump into the calculations, it's essential to understand the order of operations. This set of rules dictates the sequence in which we perform mathematical operations to ensure we arrive at the correct answer. Remember the acronym PEMDAS/BODMAS? It's our trusty guide in the world of calculations!

  • Parentheses / Brackets
  • Exponents / Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Applying this to our expression, we see that we have multiplication and subtraction. According to PEMDAS/BODMAS, we need to tackle the multiplication operations first before we even think about subtraction. This is absolutely key, guys, because messing up the order of operations can lead to a completely wrong answer. Think of it like building a house – you need to lay the foundation before you can start putting up the walls!

Taming the Multiplication Beasts: 2/5 × -3/5 and 1/13 × 3/5

Let's focus on the multiplication operations first. We have two of them: 2/5 × -3/5 and 1/13 × 3/5. Multiplying fractions is actually pretty straightforward. You simply multiply the numerators (the top numbers) and the denominators (the bottom numbers) together. Remember, a positive number multiplied by a negative number gives you a negative result. So, keep those signs in mind, folks!

For the first multiplication, 2/5 × -3/5, we multiply 2 by -3, which gives us -6. Then, we multiply 5 by 5, which gives us 25. So, 2/5 × -3/5 = -6/25. See? Not so scary after all!

Now, let's tackle the second multiplication: 1/13 × 3/5. Multiplying the numerators, 1 and 3, gives us 3. Multiplying the denominators, 13 and 5, gives us 65. Therefore, 1/13 × 3/5 = 3/65. We've conquered the multiplication beasts, guys! We're making great progress!

Rewriting the Expression: A Simpler Landscape

Now that we've handled the multiplication operations, let's rewrite our original expression with the results we've obtained. Our expression, 2/5 × -3/5 - 1/14 - 1/13 × 3/5, now transforms into -6/25 - 1/14 - 3/65. Doesn't that look a little less intimidating? We've simplified the expression, making it easier to manage. It's like decluttering your desk before tackling a big project – a clear space leads to a clearer mind!

The Subtraction Saga: Finding a Common Denominator

We're now faced with a series of subtractions: -6/25 - 1/14 - 3/65. To subtract fractions, they need to have a common denominator. This means we need to find the least common multiple (LCM) of the denominators 25, 14, and 65. Finding the LCM might seem like a daunting task, but there are methods to make it easier. You can use prime factorization or list out multiples until you find a common one. Don't worry, guys, we'll get through this together!

Let's break down the denominators into their prime factors:

  • 25 = 5 × 5
  • 14 = 2 × 7
  • 65 = 5 × 13

To find the LCM, we take the highest power of each prime factor that appears in any of the factorizations: 2, 5², 7, and 13. Multiplying these together, we get 2 × 5² × 7 × 13 = 4550. Wow, that's a big number! But don't let it scare you – we're mathematicians, and we love a good challenge!

So, the common denominator we'll use is 4550. Now, we need to convert each fraction to have this denominator. We do this by multiplying the numerator and denominator of each fraction by the appropriate factor.

Transforming the Fractions: A Common Ground

Let's convert each fraction to have the denominator 4550:

  • For -6/25, we need to multiply the denominator 25 by 182 to get 4550 (4550 / 25 = 182). So, we multiply both the numerator and denominator by 182: (-6 × 182) / (25 × 182) = -1092/4550
  • For -1/14, we need to multiply the denominator 14 by 325 to get 4550 (4550 / 14 = 325). So, we multiply both the numerator and denominator by 325: (-1 × 325) / (14 × 325) = -325/4550
  • For -3/65, we need to multiply the denominator 65 by 70 to get 4550 (4550 / 65 = 70). So, we multiply both the numerator and denominator by 70: (-3 × 70) / (65 × 70) = -210/4550

Now we have our fractions with a common denominator: -1092/4550 - 325/4550 - 210/4550. We've successfully transformed the fractions, guys! We're one step closer to the final answer.

The Grand Finale: Subtracting the Fractions

With the fractions sharing a common denominator, we can finally perform the subtraction. We simply subtract the numerators and keep the denominator the same.

So, -1092/4550 - 325/4550 - 210/4550 = (-1092 - 325 - 210) / 4550 = -1627/4550. And there you have it, folks! The solution to our mathematical adventure is -1627/4550. We did it!

Simplifying the Result: The Final Touch

While -1627/4550 is a correct answer, it's always a good idea to simplify the fraction if possible. This means finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. The GCD of 1627 and 4550 is 1. So the fraction is already in simplest form.

Conclusion: A Triumph of Mathematical Prowess

We've successfully navigated the complexities of the expression 2/5 × -3/5 - 1/14 - 1/13 × 3/5, arriving at the solution -1627/4550. We've conquered multiplication, tamed subtraction, and found a common denominator. We've even simplified our final result. This journey has demonstrated the power of the order of operations, the elegance of fraction manipulation, and the satisfaction of solving a challenging problem. Give yourselves a pat on the back, guys! You've earned it!

Remember, math isn't about memorizing formulas; it's about understanding the underlying principles and applying them with confidence. Keep practicing, keep exploring, and keep pushing your mathematical boundaries. The world of numbers is vast and fascinating, and there's always something new to discover. Until next time, happy calculating!