Algebraic Expression: Product Of 3 And Sum Of 7 And S

Hey guys! Today, we're diving deep into the awesome world of mathematics, specifically how to translate everyday phrases into cool algebraic expressions. It's like cracking a secret code, and once you get the hang of it, you'll feel like a math wizard! We're going to tackle a specific problem: "The product of 3 and the sum of 7 and s." This might sound a bit fancy, but trust me, it's totally manageable. We'll break it down, explore why certain options are right and others are wrong, and make sure you understand it inside and out. So, grab your thinking caps, and let's get this math party started!

Understanding the Core Concepts

Before we jump into solving this particular problem, let's quickly refresh some fundamental ideas. When we talk about translating phrases into algebraic expressions, we're essentially converting words into mathematical symbols. This involves understanding what different words and phrases mean in a mathematical context. For instance, words like 'sum', 'difference', 'product', and 'quotient' have specific meanings. 'Sum' means addition (+), 'difference' means subtraction (-), 'product' means multiplication (* or $\times$), and 'quotient' means division (/).

Our problem mentions "The product of 3 and the sum of 7 and s." Let's dissect this phrase. We have two main parts here: "the product of 3" and "the sum of 7 and s." The word 'and' here acts as a connector, indicating that we need to multiply these two parts together. So, the overall structure is going to be something like $3 \times (\text{the sum of 7 and s})$.

Now, let's focus on the second part: "the sum of 7 and s." As we know, 'sum' means addition. So, "the sum of 7 and s" translates directly to $7 + s$. It's crucial to remember that the order in which we add numbers doesn't change the result (commutative property), so $s + 7$ would also be correct. However, when writing expressions, we often follow a standard convention, and in this case, $7+s$ is perfectly fine.

Putting it all together, we need to find the product of 3 and $(7+s)$. This means we multiply 3 by the entire expression $(7+s)$. So, the algebraic expression representing the phrase is $3(7+s)$. The parentheses are super important here because they indicate that we are multiplying 3 by the entire sum of 7 and s, not just by 7.

This process of breaking down a phrase, identifying the mathematical operations, and using parentheses correctly is the key to mastering algebraic translations. It's all about careful reading and understanding the language of mathematics. We'll explore the given options now and see which one matches our derived expression.

Analyzing the Given Options

Alright, mathletes, we've figured out that the phrase "The product of 3 and the sum of 7 and s" should translate to $3(7+s)$. Now, let's look at the options provided and see which one matches our hard work. Remember, in mathematics, precision is key, and even a small difference can lead to a completely different answer.

Here are the options:

A. $3-7s$ B. $3(7-s)$ C. $3(7+s)$ D. $3+7s$

Let's break down each option and see if it aligns with our understanding of the original phrase:

Option A: $3-7s$

This expression involves subtraction and multiplication. The term $7s$ means 7 times $s$. So, $3-7s$ translates to "3 minus the product of 7 and $s$." This is definitely not what our original phrase described. The original phrase involved a product of two things, and one of those things was a sum. Option A has a difference between two terms, and the second term is a product, not a sum. So, Option A is incorrect.

Option B: $3(7-s)$

This expression involves the product of 3 and $(7-s)$. The part $(7-s)$ represents "the difference between 7 and $s$." Our original phrase, however, asked for "the sum of 7 and $s$." Therefore, even though this option uses multiplication and involves the number 3 and the variable $s$ along with 7, it represents the wrong operation (subtraction instead of addition) within the parentheses. So, Option B is incorrect.

Option C: $3(7+s)$

Now let's look at this gem. This expression represents the product of 3 and $(7+s)$. The part $(7+s)$ perfectly translates to "the sum of 7 and $s$." And when we take the product of 3 and this sum, we get $3(7+s)$. This perfectly matches the expression we derived from the original phrase! This is our winner, folks!

Option D: $3+7s$

This expression represents the sum of 3 and $7s$. The term $7s$ means the product of 7 and $s$. So, $3+7s$ translates to "3 plus the product of 7 and $s$." Our original phrase, however, was about the product of 3 and a sum. This option describes a sum involving 3 and a product. The operations and the grouping are completely different from what the phrase described. So, Option D is incorrect.

After carefully examining each option, it's clear that Option C is the only one that accurately represents the phrase "The product of 3 and the sum of 7 and s." It's a fantastic example of how important it is to pay attention to every word and symbol when working with mathematical expressions. Keep practicing, and you'll be translating phrases like a pro in no time!

The Importance of Parentheses

Guys, let's talk about a critical element that often trips people up in algebra: parentheses. In our problem, "The product of 3 and the sum of 7 and s," the parentheses in the correct answer, $3(7+s)$, are absolutely vital. They tell us the order of operations. According to the standard order of operations (often remembered by the acronym PEMDAS or BODMAS), operations inside parentheses are performed first.

So, in $3(7+s)$, we first calculate the sum $7+s$. Once we have that result, we then multiply it by 3. This is exactly what the phrase instructed: first find the sum, and then take the product of that sum with 3.

Let's consider what would happen if we didn't use parentheses and just wrote $3 imes 7 + s$. According to PEMDAS, multiplication comes before addition. So, we would first calculate $3 imes 7 = 21$, and then add $s$ to get $21+s$. This expression, $21+s$, represents "the sum of 21 and $s$," or equivalently, "the product of 3 and 7, added to $s$." This is clearly different from our original phrase.

Another incorrect interpretation might be something like $3 imes 7 + 3 imes s$. This uses the distributive property, which is a valid mathematical property, but it's not the direct translation of the phrase as written. The phrase implies grouping the sum first. However, if we were to expand our correct answer $3(7+s)$ using the distributive property, we would indeed get $3 imes 7 + 3 imes s$, which simplifies to $21 + 3s$. This expression means "the sum of 21 and 3 times $s$." Again, not our original phrase.

Consider the incorrect option B, $3(7-s)$. This correctly uses parentheses, but it implies the subtraction of $s$ from 7, not the addition. The phrase clearly states "the sum of 7 and $s$." So, the operation inside the parentheses must be addition.

Mastering the use of parentheses is fundamental to accurately representing mathematical relationships described in words. They ensure that the intended operations are performed in the correct sequence, preventing misinterpretations and leading to the correct solution. Always read carefully to identify which part of the phrase is being acted upon by another operation. In this case, the 'product of 3' acts upon the entirety of 'the sum of 7 and s', necessitating those crucial parentheses.

Conclusion: Mastering Phrase Translation

So, there you have it, my friends! We've successfully decoded the phrase "The product of 3 and the sum of 7 and s." By carefully breaking down the phrase, identifying the keywords for mathematical operations ('product' for multiplication, 'sum' for addition), and understanding the crucial role of parentheses, we arrived at the correct algebraic expression: $3(7+s)$. We also thoroughly analyzed why the other options were incorrect, reinforcing our understanding of how different symbolic representations lead to different mathematical meanings.

This skill of translating phrases into algebraic expressions is a cornerstone of mathematics. It's not just about solving problems; it's about learning to think logically and precisely. Whether you're in a formal math class, tackling homework, or even just trying to understand a scientific concept, being able to convert words into symbols (and vice versa) is incredibly powerful. Remember to look for keywords that indicate operations and pay close attention to how phrases are grouped – often, the grouping dictates the use of parentheses.

Practice makes perfect, as they say! The more you encounter different phrases and translate them, the more comfortable and intuitive it will become. Try creating your own phrases and translating them, or find more practice problems online. Challenge yourself! Understanding concepts like the distributive property and order of operations will further enhance your ability to manipulate and interpret these expressions. Keep exploring, keep questioning, and never be afraid to dive into the fascinating world of mathematics. You've got this!