Step-by-Step Solution For (-10) + (-4) ÷ 7
Have you ever stumbled upon a math problem that looks a bit intimidating? Don't worry, guys, we've all been there! Let's break down one of those problems today: (-10) + (-4) ÷ 7. At first glance, it might seem confusing, but with a clear, step-by-step approach, we can conquer it together. This article will guide you through the process, ensuring you not only understand the solution but also the underlying principles of mathematical operations. We'll explore the order of operations, the rules of dealing with negative numbers, and how to apply these concepts to solve this particular problem. So, grab your thinking caps, and let's dive into the world of numbers!
Decoding the Order of Operations
Before we even touch the specifics of our problem, (-10) + (-4) ÷ 7, it's crucial to understand a fundamental concept in mathematics: the order of operations. Think of it as a set of rules that dictate which calculations we perform first. Without these rules, we could end up with different answers for the same problem, leading to total chaos! The order of operations is often remembered by the acronym PEMDAS, which stands for:
- Parentheses
- Exponents
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
This acronym tells us exactly what to do and when. First, we tackle anything inside parentheses. Next, we deal with exponents (like squares and cubes). Then, we perform multiplication and division, working from left to right. Finally, we handle addition and subtraction, also from left to right. Mastering PEMDAS is like having a secret weapon in your math arsenal. It ensures that you approach every problem in a logical and consistent way, leading to accurate solutions. Now, with PEMDAS firmly in our minds, let's apply it to our problem and see how it helps us unravel the mystery.
Applying PEMDAS to Our Problem
Now that we've got PEMDAS in our toolkit, let's see how it applies to our specific problem: (-10) + (-4) ÷ 7. Looking at the expression, we can identify the operations involved: addition and division. According to PEMDAS, division comes before addition. So, the first step is to tackle the division part of the equation. This means we need to calculate (-4) ÷ 7. Remember, when dividing a negative number by a positive number, the result is negative. So, (-4) ÷ 7 will give us a negative fraction. We can express this as -4/7. Now that we've handled the division, our problem looks a little simpler: (-10) + (-4/7). The next step, according to PEMDAS, is addition. We're adding a negative fraction to a negative whole number. This is where understanding how to work with fractions and negative numbers becomes crucial. Don't worry, we'll break it down step by step. To add these numbers, we need to find a common denominator and then combine the numerators. This will give us our final answer. So, let's move on to the next section and see how we can add these numbers together.
Navigating Negative Numbers and Fractions
Working with negative numbers and fractions can sometimes feel like navigating a maze, but don't fret! Once you grasp the basic rules, it becomes much easier. Let's start with negative numbers. Think of a number line: zero is in the middle, positive numbers are to the right, and negative numbers are to the left. The further a negative number is from zero, the smaller its value. For example, -10 is smaller than -4. When we add a negative number, it's like moving to the left on the number line. Now, let's talk about fractions. A fraction represents 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. To add or subtract fractions, they need to have the same denominator. This is called finding a common denominator. We do this by finding the least common multiple (LCM) of the denominators. Once the denominators are the same, we can simply add or subtract the numerators. In our problem, we have (-10) + (-4/7). We need to express -10 as a fraction with a denominator of 7. This is where our understanding of fractions and negative numbers comes together. Let's see how we can do this in the next section.
Converting Whole Numbers to Fractions
In our quest to solve (-10) + (-4/7), we need to express the whole number -10 as a fraction with a denominator of 7. This might sound tricky, but it's actually quite simple. Remember that any whole number can be written as a fraction by placing it over 1. So, -10 can be written as -10/1. Now, to get a denominator of 7, we need to multiply both the numerator and the denominator of -10/1 by 7. This is like multiplying by 1, so it doesn't change the value of the fraction, only its appearance. When we multiply -10 by 7, we get -70. When we multiply 1 by 7, we get 7. So, -10/1 is equivalent to -70/7. Now, our problem looks like this: (-70/7) + (-4/7). We're one step closer to the solution! We have two fractions with the same denominator, which means we can finally add them together. In the next section, we'll walk through the addition process and find the final answer. So, stay with me, guys, we're almost there!
Adding Fractions with Common Denominators
We've reached the point where we need to add two fractions with common denominators: (-70/7) + (-4/7). This is the easiest part of the process! When fractions have the same denominator, we simply add the numerators and keep the denominator the same. So, we add -70 and -4. Remember the rules for adding negative numbers: when you add two negative numbers, you add their absolute values and keep the negative sign. The absolute value of -70 is 70, and the absolute value of -4 is 4. Adding 70 and 4 gives us 74. Since both numbers were negative, our result is -74. Therefore, (-70) + (-4) = -74. Now, we put this result over our common denominator, which is 7. So, (-70/7) + (-4/7) = -74/7. This is our final answer! We've successfully navigated the problem and found the solution. But let's take it one step further and express this improper fraction as a mixed number. This will give us a better sense of the value of our answer.
Converting Improper Fractions to Mixed Numbers
Our final answer is currently expressed as an improper fraction: -74/7. An improper fraction is one where the numerator is greater than the denominator. To get a better grasp of the value, we can convert it to a mixed number. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator). To convert -74/7 to a mixed number, we need to divide the numerator (74) by the denominator (7). We're essentially asking: how many times does 7 fit into 74? 7 goes into 74 ten times (7 x 10 = 70). This gives us our whole number part: -10. We have a remainder of 4 (74 - 70 = 4). This remainder becomes the numerator of our fractional part, and we keep the same denominator (7). So, our fractional part is 4/7. Combining the whole number and the fractional part, we get the mixed number -10 4/7. This is the same value as -74/7, just expressed in a different way. Now, we have a clear picture of our final answer. We've not only solved the problem but also understood the underlying concepts and how to express the answer in different forms. Great job, guys!
Final Answer and Reflection
So, after carefully navigating through the order of operations, negative numbers, fractions, and conversions, we've arrived at the final answer to our problem: (-10) + (-4) ÷ 7 = -74/7 or -10 4/7. That's quite an accomplishment! But more than just getting the right answer, it's important to reflect on the process we used. We started by understanding PEMDAS, which gave us the roadmap for solving the problem. We then tackled the division part, followed by converting the whole number to a fraction, and finally, adding the fractions with common denominators. We even went the extra mile and converted our improper fraction to a mixed number. By breaking down the problem into smaller, manageable steps, we made it much less daunting. This approach can be applied to any math problem, no matter how complex it may seem. Remember, math is like a puzzle. Each piece has its place, and by understanding the rules and principles, we can fit them together to reveal the solution. So, keep practicing, keep exploring, and most importantly, keep asking questions! The world of mathematics is full of fascinating discoveries waiting to be made. You guys got this!
