Math Problem: Sum, Difference, And More!
Hey guys! Let's dive into a fun math problem that involves sums, differences, and a little bit of everything in between. We're going to break down a word problem that might seem tricky at first, but trust me, it's totally manageable once we take it step by step. Our mission is to understand what "one half of the sum of a number and 10, plus the difference of m and 7 increased by 15" really means. So, grab your pencils, and let's get started!
Understanding the Basics
Before we jump into the main problem, let's quickly brush up on some key mathematical terms. This will help us translate the words into math symbols and equations. Remember, math is just another language, and once you learn the vocabulary, you can understand almost anything!
- Sum: This is the result of adding two or more numbers together. For example, the sum of 5 and 3 is 8.
- Difference: This is the result of subtracting one number from another. For example, the difference between 10 and 4 is 6.
- Variable: A variable is a symbol, usually a letter, that represents an unknown number. In our problem, we have the variables "a number" and "m."
- Increased by: This means we need to add a certain amount to a number. For example, "5 increased by 2" means 5 + 2, which equals 7.
- One half of: This means we need to divide something by 2 or multiply it by 1/2. For example, "one half of 10" is 10 / 2 or (1/2) * 10, which equals 5.
With these basics in mind, we're ready to tackle our word problem. Let's break it down bit by bit.
Breaking Down the Problem Step-by-Step
Okay, let's dissect this problem piece by piece. Our goal is to turn the words into a mathematical expression. Remember, the key to solving word problems is to take it slow and translate each part carefully.
1. "One half of the sum of a number and 10"
Let's start with the first part: "one half of the sum of a number and 10." The first thing we need to do is represent "a number" with a variable. Let's use the variable x. So, "the sum of a number and 10" can be written as x + 10.
Now, we need to find “one half of” this sum. As we discussed earlier, “one half of” means we need to multiply the sum by 1/2 or divide it by 2. So, “one half of the sum of a number and 10” can be written as:
(1/2) * (x + 10) or (x + 10) / 2
We'll stick with the first form, (1/2) * (x + 10), for now. Remember, the parentheses are important here because they tell us to add x and 10 before multiplying by 1/2.
2. "The difference of m and 7"
Next up, we have "the difference of m and 7." This part is a bit more straightforward. The word "difference" tells us we need to subtract. So, "the difference of m and 7" can be written as:
m - 7
Here, m is another variable, and we're subtracting 7 from it.
3. "Increased by 15"
Now, let's look at "increased by 15." This means we need to add 15 to something. But what are we adding 15 to? Well, we're adding 15 to "the difference of m and 7," which we already know is m - 7. So, "the difference of m and 7 increased by 15" can be written as:
(m - 7) + 15
4. Putting It All Together
Okay, we've broken down each part of the problem. Now, let's put it all together. The original problem was:
"One half of the sum of a number and 10, plus the difference of m and 7 increased by 15"
We know that:
- "One half of the sum of a number and 10" is (1/2) * (x + 10)
- "The difference of m and 7 increased by 15" is (m - 7) + 15
The word "plus" tells us to add these two expressions together. So, the complete mathematical expression is:
(1/2) * (x + 10) + (m - 7) + 15
Simplifying the Expression
We've successfully translated the word problem into a mathematical expression! Give yourselves a pat on the back. But we're not quite done yet. It's always a good idea to simplify expressions if we can. Let's see what we can do with our expression:
(1/2) * (x + 10) + (m - 7) + 15
1. Distribute the 1/2
First, let's distribute the 1/2 across the parentheses in the first term. This means we'll multiply both x and 10 by 1/2:
(1/2) * x = x/2
(1/2) * 10 = 5
So, (1/2) * (x + 10) becomes x/2 + 5. Our expression now looks like this:
x/2 + 5 + (m - 7) + 15
2. Combine Like Terms
Next, let's look for any like terms we can combine. Like terms are terms that have the same variable or are just constants (numbers without variables). In our expression, we have the constants 5, -7, and 15. Let's combine them:
5 - 7 + 15 = 13
So, our expression simplifies to:
x/2 + m + 13
The Final Answer
And there you have it! We've simplified the expression as much as we can. The final answer is:
x/2 + m + 13*
This expression represents "one half of the sum of a number and 10, plus the difference of m and 7 increased by 15." We've successfully translated a complex word problem into a concise mathematical expression.
Why Is This Important?
You might be wondering, "Why do we need to do this?" Well, translating word problems into mathematical expressions is a crucial skill in algebra and beyond. It allows us to take real-world situations and represent them in a way that we can solve using mathematical tools.
For example, you might encounter similar problems when calculating costs, planning projects, or analyzing data. The ability to break down a problem, identify the key information, and translate it into an equation is a valuable skill in many areas of life.
Practice Makes Perfect
The best way to get better at these types of problems is to practice. Try working through similar word problems, breaking them down step by step, and translating them into mathematical expressions. Don't be afraid to make mistakes – that's how we learn! And remember, math is like a muscle; the more you use it, the stronger it gets.
Conclusion
So, guys, we've tackled a pretty challenging math problem today, and I'm super proud of you for sticking with it. We've learned how to break down a complex word problem, translate it into a mathematical expression, and simplify it. Remember, the key is to take it step by step, identify the key information, and use your math vocabulary.
Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!