Solving 1/8 + (-7/16) + (0.8)(1/4): A Math Guide

by Andrew McMorgan 49 views

Hey math enthusiasts! Ever stumbled upon a problem that looks like a jumble of fractions and decimals? Today, we're going to break down one such puzzle: 1/8 + (-7/16) + (0.8)(1/4). Don't worry, it's not as intimidating as it looks. We'll go through each step together, so you'll not only understand the solution but also the why behind it. Think of it as a mini math adventure – let's dive in!

Understanding the Order of Operations

Before we even think about adding fractions or dealing with decimals, we need to remember our trusty friend, the order of operations, often remembered by the acronym PEMDAS (or BODMAS, depending on where you learned math!). This tells us the sequence in which we need to perform operations:

  1. Parentheses (or Brackets)
  2. Exponents (or Orders)
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

In our problem, we have addition, subtraction (due to the negative fraction), and multiplication. According to PEMDAS, we tackle multiplication first. This is a crucial step! Getting the order wrong can throw off the entire solution. We need to make sure we're following this order to get the right answer, guys.

Step 1: Tackling the Multiplication

Our equation has a multiplication part: (0.8)(1/4). To make things easier, let's convert the decimal 0.8 into a fraction. We know that 0.8 is the same as 8/10. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us 4/5. So now our multiplication looks like this: (4/5)(1/4).

Multiplying fractions is pretty straightforward: we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, 4/5 multiplied by 1/4 is (4 * 1) / (5 * 4) = 4/20. Now, we can simplify this fraction again. Both 4 and 20 are divisible by 4, so we divide both by 4, resulting in 1/5. So, (0.8)(1/4) simplifies to 1/5. See? Not so scary when we break it down!

Step 2: Rewriting the Equation

Now that we've conquered the multiplication, let's rewrite our original equation with the simplified term. Our equation now looks like this: 1/8 + (-7/16) + 1/5. We've transformed our problem into one involving only addition and subtraction of fractions. This is progress, guys! We're one step closer to the final answer. This step is important because it makes the next calculations much easier to manage. By simplifying the multiplication first, we've set ourselves up for a smoother ride.

Step 3: Finding a Common Denominator

To add or subtract fractions, they need to have the same denominator (the bottom number). This is like making sure we're adding apples to apples, not apples to oranges. To find the common denominator for 1/8, -7/16, and 1/5, we need to find the least common multiple (LCM) of 8, 16, and 5.

Let's list the multiples of each number:

  • Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...
  • Multiples of 16: 16, 32, 48, 64, 80...
  • Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80...

The smallest number that appears in all three lists is 80. So, 80 is our least common multiple and will be the common denominator for our fractions. This is a key concept in fraction arithmetic. Without a common denominator, we can't accurately add or subtract the fractions. It's like trying to add different units, like centimeters and inches – we need to convert them to the same unit first.

Step 4: Converting the Fractions

Now we need to convert each fraction to have a denominator of 80. To do this, we'll multiply the numerator and denominator of each fraction by the number that makes the denominator 80:

  • For 1/8: We need to multiply 8 by 10 to get 80, so we multiply both the numerator and denominator by 10: (1 * 10) / (8 * 10) = 10/80.
  • For -7/16: We need to multiply 16 by 5 to get 80, so we multiply both the numerator and denominator by 5: (-7 * 5) / (16 * 5) = -35/80.
  • For 1/5: We need to multiply 5 by 16 to get 80, so we multiply both the numerator and denominator by 16: (1 * 16) / (5 * 16) = 16/80.

Now our equation looks like this: 10/80 + (-35/80) + 16/80. See how much cleaner it looks? We're dealing with fractions that have the same denominator, making the next step a breeze. This conversion is super important because it allows us to perform the addition and subtraction accurately. It's like putting all the pieces of a puzzle into the right format before we start assembling it.

Step 5: Adding and Subtracting the Fractions

Now that our fractions have a common denominator, we can simply add and subtract the numerators. Remember, we're adding the numerators while keeping the denominator the same. So, we have:

10/80 + (-35/80) + 16/80 = (10 + (-35) + 16) / 80

Let's break this down step by step:

  • 10 + (-35) = -25
  • -25 + 16 = -9

So, our result is -9/80. This is the simplified form of our fraction because 9 and 80 don't share any common factors other than 1. We've successfully added and subtracted the fractions! This is the heart of the problem, where all our previous work comes together to give us the answer. It's like the final brushstroke on a painting.

Final Answer

Therefore, 1/8 + (-7/16) + (0.8)(1/4) = -9/80. And there you have it! We've solved the problem step by step, from understanding the order of operations to finding common denominators and finally adding and subtracting the fractions. Hopefully, you feel a bit more confident tackling similar problems now.

Key Takeaways

  • Order of Operations (PEMDAS/BODMAS): Always remember the order! Multiplication and division come before addition and subtraction.
  • Converting Decimals to Fractions: This can often simplify calculations.
  • Finding a Common Denominator: Essential for adding and subtracting fractions.
  • Simplifying Fractions: Always reduce your answer to its simplest form.

Math can seem daunting at first, but by breaking it down into smaller steps, even complex problems become manageable. Keep practicing, guys, and you'll be math whizzes in no time! And remember, it's okay to make mistakes – that's how we learn. The key is to keep trying and keep exploring. Happy calculating!