Algebraic Expressions: Ten Times The Sum Of Seven And A Number

Hey there, math enthusiasts! Welcome back to Plastik Magazine, where we break down those tricky math concepts so they’re as clear as day. Today, we're diving into the world of algebraic expressions, specifically tackling this head-scratcher: "Write and evaluate the expression: ten times the sum of seven and a number; evaluate when $j=1$." Sounds a bit wordy, right? But don't sweat it, guys. We'll unpack this step-by-step, making sure you're not just getting the answer, but really understanding it. We'll explore why certain expressions work and others don't, and how to confidently solve these kinds of problems every single time. So, grab your notebooks (or just your brilliant brains), and let's get started on mastering this algebraic puzzle!

Understanding the Wording: Breaking Down the Problem

Alright, let's get straight to the heart of it. The phrase "ten times the sum of seven and a number" is where we need to focus our energy first. When we see "ten times", in the language of math, that usually means multiplication. So, we know we're going to be multiplying something by 10. What are we multiplying? We're multiplying "the sum of seven and a number." Now, "the sum of seven and a number" tells us we need to add two things together: the number 7, and an unknown quantity. In algebra, we use letters, often called variables, to represent these unknown numbers. The problem tells us this number is represented by the variable $j$. So, the phrase "the sum of seven and a number" translates directly to $7 + j$. Putting it all together, "ten times the sum of seven and a number" becomes $10 imes (7 + j)$. It's super important to use parentheses here, guys. Why? Because we need to find the sum first, and then multiply that entire sum by ten. If we wrote $10 imes 7 + j$, it would mean multiply 10 by 7, and then add $j$, which isn't what the original phrase asked for. Always remember, when a phrase uses words like "sum" or "difference" before specifying multiplication or division, parentheses are usually your best friends to ensure the operation is done in the correct order.

Now, the problem also gives us a specific value for $j$. It says, "evaluate when $j=1$." This means we take our algebraic expression, $10(7+j)$, and wherever we see $j$, we replace it with the number 1. So, our expression becomes $10(7+1)$. To evaluate this, we follow the order of operations (PEMDAS/BODMAS, remember that?). First, we handle what's inside the parentheses: $7 + 1 = 8$. Then, we perform the multiplication: $10 imes 8 = 80$. So, the value of the expression when $j=1$ is 80. This is a key step: writing the expression correctly is half the battle, and evaluating it accurately is the other half. We've now successfully translated the words into a mathematical form and found its value for a specific variable. Pretty neat, huh?

Evaluating the Options: Finding the Correct Answer

Okay, so we've figured out that the correct expression is $10(7+j)$ and its value when $j=1$ is 80. Now, let's look at the options provided to see which one matches our findings. This is where careful comparison comes in, guys. We need to check both the expression and the evaluated value for each option.

• Option A: $10 imes 7+j$; when $j=1$, the value is 71 Let's evaluate the expression $10 imes 7+j$ when $j=1$. Following the order of operations, we multiply first: $10 imes 7 = 70$. Then we add $j$: $70 + 1 = 71$. The evaluated value is correct (71), but the expression itself, $10 imes 7+j$, does not correctly represent "ten times the sum of seven and a number." It represents "the sum of seventy and a number." So, this option gets the evaluation right based on its own expression, but the expression is wrong. We're looking for the expression that matches the description. Therefore, Option A is incorrect.

• Option B: $7(10+j)$; when $j=1$, the value is 77 Let's check the expression $7(10+j)$. This means 7 multiplied by the sum of 10 and $j$. This doesn't match our original wording at all. If we evaluate it with $j=1$, we get $7(10+1) = 7(11) = 77$. The evaluation is correct for this specific expression, but the expression itself is fundamentally different from what we needed to write. So, Option B is also incorrect.

• Option C: $10(7-j)$; when $j=1$, the value is 60 Here we have the expression $10(7-j)$. This translates to "ten times the difference of seven and a number." The word "difference" means subtraction, not addition ("sum"). So, the expression is incorrect from the start. If we were to evaluate it with $j=1$, we'd get $10(7-1) = 10(6) = 60$. Again, the evaluation matches the expression, but the expression doesn't match the problem statement. Option C is a no-go.

• Option D: $10(7+j)$; when $j=1$, the value is 80 Let's look at this one carefully. The expression is $10(7+j)$. This perfectly matches our breakdown: ten times (the 10 multiplied by) the sum of seven and a number (the $7+j$ inside the parentheses). Now, let's evaluate it when $j=1$. We substitute $j$ with 1: $10(7+1)$. Following the order of operations, we first calculate the sum inside the parentheses: $7+1 = 8$. Then, we multiply: $10 imes 8 = 80$. So, the value is indeed 80. This option gets both the expression and the evaluated value correct! Bingo!

The Correct Answer and Why It Works

So, after carefully analyzing each option, we've confirmed that Option D is the correct answer. It correctly translates the word problem into the algebraic expression $10(7+j)$ and accurately evaluates it to 80 when $j=1$. The key takeaway here, guys, is the importance of translating words into mathematical symbols accurately. Phrases like "sum of," "difference of," "product of," and "quotient of" dictate which operations to perform, and often, parentheses are crucial to ensure the correct order of operations. Remember, "ten times the sum of seven and a number" means you first find the sum ($7+j$) and then multiply the result by ten ($10 imes (7+j)$). It's not "ten times seven, plus a number" ($10 imes 7 + j$), nor is it "seven times the sum of ten and a number" ($7 imes (10+j)$). Precision in language is just as important in math as it is in everyday life!

Let's quickly revisit why the other options were incorrect. Option A had the correct evaluated value for its own expression, but its expression, $10 imes 7+j$, represented "the sum of seventy and a number," not "ten times the sum of seven and a number." Option B, $7(10+j)$, incorrectly represented the problem, meaning "seven times the sum of ten and a number." Option C, $10(7-j)$, used subtraction instead of addition, translating to "ten times the difference of seven and a number." Each of these options failed to capture the specific wording of the original problem, even if some of them had a correct numerical result for their own incorrect expressions. This highlights the importance of not just getting a number, but ensuring the entire mathematical structure is sound and accurately reflects the problem statement. So, next time you see a word problem, take a deep breath, break it down phrase by phrase, and don't forget those parentheses when they're needed. You've got this!

Final Thoughts: Mastering Algebraic Expressions

We've successfully navigated the complexities of writing and evaluating algebraic expressions today. It's clear that understanding the nuances of mathematical language is paramount. When faced with a problem like "ten times the sum of seven and a number," the translation to $10(7+j)$ is the critical first step. This involves recognizing that "times" signifies multiplication and "sum" signifies addition, and crucially, that the addition must be performed before the multiplication due to the phrasing, hence the necessity of parentheses. Evaluating this expression with $j=1$ led us to $10(7+1) = 10(8) = 80$. This meticulous approach ensures accuracy and builds a solid foundation for more advanced algebraic concepts. It's not just about memorizing rules, guys; it's about understanding the logic behind them.

We saw how small changes in wording lead to vastly different expressions and, consequently, different answers. Option A, $10 imes 7+j$, would mean $70+j$, which is significantly different from $10(7+j)$. Option B, $7(10+j)$, shifts the multiplier and the number being added. Option C, $10(7-j)$, changes the operation from addition to subtraction. Each incorrect option serves as a valuable lesson in precision. The goal isn't just to find an answer, but to find the correct answer that precisely matches the problem's conditions. This reinforces the idea that in mathematics, clarity and accuracy in representation are non-negotiable. By consistently breaking down word problems, identifying keywords, and applying the order of operations correctly, you can confidently tackle any algebraic expression challenge that comes your way. Keep practicing, keep questioning, and you'll become a math whiz in no time! Remember, every problem solved is a step towards mastery. Don't hesitate to re-read, re-evaluate, and ensure your mathematical sentences make as much sense as the English sentences they represent. That's the power of algebra!