Algebraic Phrases: Translate '12 Times The Sum Of A Number And 7/10'

Hey mathletes, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of algebraic phrases. You know, those wordy problems that can sometimes look like a secret code? Well, guess what? We're here to crack that code and make it super easy for you guys to understand. Our main mission today is to figure out exactly how to represent the phrase 'twelve times the sum of a number and seven-tenths' in algebraic form. This isn't just about getting the right answer; it's about building that solid foundation in understanding how words translate into the elegant language of mathematics. We'll break down the phrase piece by piece, explore why certain operations are used, and ultimately arrive at the correct algebraic expression. So, buckle up, grab your favorite thinking cap, and let's get this algebraic adventure started! We'll be looking at the different options provided and dissecting why only one truly captures the essence of the given phrase. It’s all about precision and understanding the order of operations when we’re dealing with these kinds of problems.

Deconstructing the Phrase: 'Twelve Times the Sum of a Number and Seven-Tenths'

Alright guys, let's get down and dirty with the phrase itself: 'twelve times the sum of a number and seven-tenths'. To translate this into an algebraic expression, we need to break it down word by word and understand what each part signifies mathematically. First off, we have 'a number'. In algebra, when we see 'a number' or 'some number', we typically represent it with a variable. Let's use '$y$' for our number, just like in the options provided. Now, what about the 'sum'? The word 'sum' tells us we need to perform an addition operation. And what are we adding? We're adding our number ('$y$') and 'seven-tenths'. Seven-tenths is a fraction, which we can write as $\frac{7}{10}$. So, the 'sum of a number and seven-tenths' translates to $y + \frac{7}{10}$.

Now, here's the crucial part: 'twelve times'. The phrase 'twelve times' indicates multiplication. But what are we multiplying by twelve? Are we multiplying just the number '$y$' by twelve, or are we multiplying the entire sum we just figured out ($y + \frac{7}{10}$)? The phrasing 'twelve times the sum of' is key here. It tells us that the multiplication by twelve applies to the result of the addition. This means we need to group the sum using parentheses. Therefore, we take twelve and multiply it by the entire expression $(y + \frac{7}{10})$. This leads us to the algebraic expression $12 \left(y + \frac{7}{10}\right)$. This expression accurately represents the original phrase by ensuring that the addition happens before the multiplication, as dictated by the word 'sum' being acted upon by 'twelve times'. It’s a subtle but critical distinction in algebraic translation.

Analyzing the Options: Spotting the Correct Representation

Now that we've carefully broken down the phrase and built our algebraic expression, let's look at the options given and see which one matches our findings. Remember, we're looking for $12 \left(y + \frac{7}{10}\right)$.

• Option A: $12 y+\frac{7}{10}$ This expression translates to 'twelve times a number, plus seven-tenths'. Notice that the addition of $\frac{7}{10}$ is separate from the multiplication of $12y$. This means the 'sum' part isn't being multiplied by twelve. The phrase would have to be something like 'twelve times a number, and then add seven-tenths' for this to be correct. So, A is out.

• Option B: $12\left(y+\frac{7}{10}\right)$ This expression translates to 'twelve times the quantity (or sum) of a number and seven-tenths'. The parentheses here are super important! They tell us to perform the operation inside the parentheses first – that's the sum of '$y$' and '$\frac{7}{10}$'. Then, we multiply the result by twelve. This perfectly matches our breakdown of the original phrase. Bingo!

• Option C: $\frac{7}{10} y+12$ This expression translates to 'seven-tenths times a number, plus twelve'. This is completely different from our original phrase. The roles of twelve and seven-tenths are swapped, and the operation associated with twelve is now addition. So, C is definitely not the one.

• Option D: $y\left(\frac{7}{10}+12\right)$ This expression translates to 'a number times the sum of seven-tenths and twelve'. While it involves multiplication and addition, the structure is wrong. It suggests multiplying '$y$' by the sum of $\frac{7}{10}$ and 12, which isn't what our original phrase states. Our phrase says 'twelve times the sum', not 'a number times the sum'.

See how important those little words and the placement of parentheses are, guys? Option B is the only one that correctly captures the meaning of 'twelve times the sum of a number and seven-tenths'.

The Importance of Parentheses in Algebraic Expressions

Let's really hammer home why parentheses are the MVPs (Most Valuable Players) when translating algebraic phrases. When we talk about the 'sum of a number and seven-tenths', we're essentially creating a single entity: the result of $y + \frac{7}{10}$. When the phrase then says 'twelve times' this sum, it means we're applying the multiplication to that entire entity. Parentheses act like a signal, telling us: "Hey, do what's inside me first!" In our case, it tells us to calculate $y + \frac{7}{10}$ before we multiply by 12.

Without parentheses, the order of operations (PEMDAS/BODMAS) would dictate that multiplication comes before addition. So, if we wrote $12y + \frac{7}{10}$ (Option A), it would mean multiply 12 by '$y$' first, and then add $\frac{7}{10}$. This is a completely different mathematical operation and outcome. Imagine you have 12 apples, and you want to give them to a group of friends plus an extra friend. If the phrase was 'twelve times the number of friends, plus one more friend', you'd calculate the total number of friends first, then multiply by 12, and then maybe add that extra friend. But if it's 'twelve times the sum of friends and one extra', you are grouping the friends and the extra person together before multiplying by 12. It’s a subtle difference in wording that leads to a massive difference in the math.

Consider the phrase: 'the sum of twelve times a number and seven-tenths'. This would translate to $12y + \frac{7}{10}$. See the difference? The word 'sum' comes after 'twelve times a number', indicating that the addition is the final step. In our original problem, 'twelve times' comes before 'the sum', indicating that the multiplication envelops the sum. This is why Option B: $12\left(y+\frac{7}{10}\right)$ is the only accurate representation. It correctly groups the addition using parentheses, ensuring that the sum is calculated before the multiplication by twelve.

Conclusion: Mastering Algebraic Translation

So, there you have it, folks! Translating algebraic phrases is all about careful reading and understanding the precise meaning of mathematical terms and their order. We've successfully broken down 'twelve times the sum of a number and seven-tenths' and confirmed that the correct algebraic expression is Option B: $12\left(y+\frac{7}{10}\right)$. This process reinforces the critical role of understanding vocabulary like 'sum', 'times', 'difference', and 'quotient', as well as the power of parentheses in defining the order of operations. By mastering these skills, you're not just solving problems; you're learning to speak the universal language of mathematics fluently. Keep practicing, keep questioning, and remember that every complex algebraic expression is just a series of simpler terms waiting to be understood. You guys are doing great, and with a little more practice, you'll be algebraic wizards in no time! Stay tuned to Plastik Magazine for more math adventures!