Solving Twice The Sum Of A Number And 4.5 Is 3 More Than 8

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Hey there, math enthusiasts! Let's dive into a fun little puzzle that combines the worlds of algebra and word problems. We're going to break down the statement: "Twice the sum of a number and 4.5 is 3 more than 8," translate it into a symbolic representation, and then, for the grand finale, solve it! Buckle up, because we're about to embark on a mathematical adventure.

Decoding the Statement: From Words to Symbols

Before we jump into the options, let's dissect the given statement piece by piece. This is crucial, guys, because translating words into mathematical expressions is the first step in solving any word problem. So, grab your thinking caps and let's get started!

1. "A Number"

Okay, so we have "a number." In algebra, we often represent unknown numbers with variables. Let's use the classic 'n' for our number. You could use any letter, really – x, y, z – but 'n' feels right in this case, right?

2. "The Sum of a Number and 4.5"

Next up, we have "the sum of a number and 4.5." Sum, in math lingo, means addition. So, we're adding our number (n) and 4.5. This translates to n + 4.5. Simple enough, isn't it?

3. "Twice the Sum"

Now, things get a little more interesting. We have "twice the sum." Twice means multiplying by 2. But we're not just multiplying n or 4.5 by 2; we're multiplying the entire sum (n + 4.5) by 2. To show this, we use parentheses: 2(n + 4.5). The parentheses make sure we're treating (n + 4.5) as a single unit before multiplying by 2.

4. "Is"

This is a big one! "Is" is often the most important word in these problems because it translates directly to the equals sign (=). This is where we start to build our equation. So, everything we've translated so far goes on the left side of the equals sign: 2(n + 4.5) = ...

5. "3 More Than 8"

Last but not least, we have "3 more than 8." This means we're adding 3 to 8. So, 3 more than 8 is simply 8 + 3, which equals 11. This goes on the right side of our equation.

Putting It All Together

Now, let's combine all the pieces we've decoded. We have: 2(n + 4.5) = 8 + 3. Or, simplified, 2(n + 4.5) = 11. This is the symbolic representation of our original statement.

Evaluating the Options

Now that we've translated the statement ourselves, let's take a look at the options and see which one matches our masterpiece:

  • A. 2(n+4.5)+3-815 – This one is a bit of a mess, isn't it? It has an extra +3-815 tacked on, which doesn't make sense in the context of our original statement. So, option A is a definite no-go.
  • B. 2(n+4.5)=8+311 – This looks promising! It has the 2(n + 4.5) on the left side, which we know is correct. And it has 8 + 3, which represents "3 more than 8." The only thing is, 8 + 311 is wrong because 8 + 3 is just 11. The 11 at the end is incorrect here so this option is wrong.
  • C. 2 n+(4.5+3)=8: 11 – This one messes up the order of operations. It only multiplies n by 2 and adds 3 to 4.5 inside the parentheses. Remember, we need to multiply the entire sum of n and 4.5 by 2. So, this option is incorrect.
  • D. 2(n+4.5)=8+3 – Bingo! This is the winner! It perfectly matches our translated equation: 2(n + 4.5) = 11. Option D correctly represents the statement "Twice the sum of a number and 4.5 is 3 more than 8." This one is correct.

Cracking the Code: Solving the Equation

We've successfully translated the words into symbols, and we've identified the correct representation. Now comes the exciting part: solving for n! This is where we put our algebra skills to the test.

1. Distribute the 2

Our equation is 2(n + 4.5) = 11. To get rid of the parentheses, we need to distribute the 2. This means we multiply both n and 4.5 by 2:
2 * n = 2n 2 * 4.5 = 9 So, our equation becomes 2n + 9 = 11. We're one step closer!

2. Isolate the Variable Term

Our goal is to get n by itself on one side of the equation. To do this, we need to get rid of the +9. We can do this by subtracting 9 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced:
2n + 9 - 9 = 11 - 9 This simplifies to 2n = 2. Looking good!

3. Solve for n

We're almost there! Now we have 2n = 2. To get n by itself, we need to divide both sides of the equation by 2:
(2n) / 2 = 2 / 2 This gives us n = 1. Woo-hoo! We've found our solution.

The Grand Finale: The Solution

So, after our mathematical journey, we've arrived at the answer. The solution to the equation 2(n + 4.5) = 11 is n = 1. That means the number that satisfies the original statement, "Twice the sum of a number and 4.5 is 3 more than 8," is 1. Awesome, right?

Why This Matters: The Power of Algebra

Now, you might be thinking, "Okay, that was a fun puzzle, but why does this matter?" Well, guys, this is the essence of algebra! We're taking real-world situations, translating them into mathematical language, and using equations to solve for unknowns. This is a skill that's used in countless fields, from science and engineering to finance and economics. Being able to translate words into symbols and solve equations is a superpower in the world of problem-solving.

Key Takeaways

Let's recap the key things we learned in this adventure:

  • Translating Words to Symbols: Break down the statement piece by piece, identify key words like "sum," "twice," and "is," and translate them into mathematical symbols and operations.
  • The Importance of Parentheses: Use parentheses to group terms and ensure the correct order of operations.
  • Solving Equations: Use inverse operations (addition/subtraction, multiplication/division) to isolate the variable and solve for the unknown.
  • Checking Your Work: After solving, plug your solution back into the original equation to make sure it works.

Practice Makes Perfect

Like any skill, mastering algebra takes practice. So, don't be afraid to tackle more word problems, translate more statements, and solve more equations. The more you practice, the more confident you'll become in your algebraic abilities.

So, there you have it! We've successfully unraveled the mystery of "Twice the sum of a number and 4.5 is 3 more than 8." We translated it into symbols, identified the correct representation, and solved for the unknown. You're now one step closer to becoming a math whiz!

Keep practicing, keep exploring, and keep having fun with math! You've got this!