Solve: Ten Less Than Twice A Number

Hey guys! Today we're diving deep into the awesome world of algebra, specifically tackling a word problem that can be a bit tricky if you don't break it down. We're going to figure out which equation correctly represents the sentence: "ten less than twice a number, $n$, is sixteen." This is super important because understanding how to translate words into mathematical expressions is like having a secret code for solving all sorts of problems. So, grab your notebooks, and let's get this math party started! We'll go through each option and see why only one truly captures the meaning of the original statement. It’s all about paying attention to those little words that change everything.

Understanding the Phrases

Alright, let's break down this phrase piece by piece. The sentence is: "ten less than twice a number, $n$, is sixteen." First off, we have "twice a number, $n$." In algebra, when we talk about "twice a number," it means we're multiplying that number by 2. So, "twice a number, $n$" translates directly to $2n$. This is a pretty straightforward translation. Now, let's look at the next part: "ten less than..." This is where things can get a little confusing for some folks. When you see "less than," it often means subtraction, but the order is crucial! "Ten less than something" means you're taking that 'something' and then subtracting 10 from it. So, if it's "ten less than twice a number," it's not $10 - 2n$; it's actually $2n - 10$. Think of it this way: if you have $50, and I tell you I have ten less than that, you wouldn't say $10 - 50$ (which is negative $40!), you'd say $50 - 10 = 40$. So, the order matters! The phrase "ten less than twice a number, $n$" therefore translates to $2n - 10$. Keep this in mind, it's a common tripping point, but once you get it, it’s smooth sailing!

Decoding the Full Equation

Now that we've nailed down "ten less than twice a number, $n$" as $2n - 10$, let's look at the rest of the sentence: "...is sixteen." In math language, the word "is" almost always means equals. So, "is sixteen" translates to $= 16$. Putting it all together, the complete equation that represents the sentence "ten less than twice a number, $n$, is sixteen" is $2n - 10 = 16$. This equation perfectly captures the relationship described in the words. It states that when you take twice the number $n$ and then subtract 10 from it, the result is 16. This is the core of translating word problems into algebra; you dissect the sentence, understand the meaning of each phrase, and then assemble the mathematical symbols. It’s like solving a puzzle, but instead of a picture, you’re building a solvable equation.

Analyzing the Options

Let's take a look at the options provided and see which one matches our derived equation. We have:

A. $10-2+n=16$ B. $10+2-n=16$ C. $10-2n=16$ D. $2n-10=16$

Based on our breakdown, we found that the correct equation should be $2n - 10 = 16$. Now, let's compare this to the options. Option A, $10-2+n=16$, is incorrect because it interprets "ten less than" as $10$ minus something and also adds $n$ instead of multiplying it by 2. Option B, $10+2-n=16$, is also incorrect for similar reasons; it changes the operations and the role of $n$. Option C, $10-2n=16$, gets the "twice a number" part as $2n$, but it incorrectly interprets "ten less than" as $10$ minus $2n$. Remember our rule: "less than" means the first number is subtracted from the second. Finally, we have Option D: $2n-10=16$. This matches exactly with the equation we built step-by-step: "twice a number, $n$" ($2n$) followed by "ten less than" ($-10$) and "is sixteen" ($=16$). So, the correct choice is D.

Solving the Equation (Bonus Round!)

While the question only asked for the equivalent equation, let's go the extra mile and solve for $n$ just for fun! We have the equation $2n - 10 = 16$. Our goal is to isolate $n$. First, we want to get the term with $n$ by itself, so we'll add 10 to both sides of the equation to cancel out the $-10$:

$2n - 10 + 10 = 16 + 10$

$2n = 26$

Now, to get $n$ all by itself, we need to undo the multiplication by 2. We do this by dividing both sides by 2:

rac{2n}{2} = rac{26}{2}

$n = 13$

So, the number $n$ is 13. Let's check our answer by plugging it back into the original sentence: "ten less than twice a number, $n$, is sixteen." Twice the number 13 is $2 imes 13 = 26$. Ten less than 26 is $26 - 10 = 16$. And yes, 16 is indeed 16! Our equation and our solution are correct. This bonus step really solidifies that we understood the problem and translated it perfectly.

Conclusion: Mastering Translation

So there you have it, guys! We've successfully translated a tricky word problem into a neat algebraic equation and even solved it. The key takeaway here is the importance of carefully dissecting word problems. Pay close attention to phrases like "less than," "more than," "times," and "is," as they directly correspond to mathematical operations and the structure of your equation. Always remember that "a less than b" translates to $b-a$, not $a-b$. Mastering this translation skill is fundamental to your success in mathematics. It allows you to tackle complex problems by breaking them down into manageable steps. Keep practicing, and you'll become a word problem whiz in no time! Remember, the journey of a thousand mathematical miles begins with a single translated phrase. Keep your thinking caps on, and happy problem-solving!