Equation For A Number Problem: 3n - 15 + N = 101

Let's break down this math problem step by step, guys, and find the correct equation. We're given a scenario where a number, represented by n, is added to a modified version of itself. This modified version is "15 less than 3 times itself." The final result of this addition is 101. Our mission is to pinpoint the equation that accurately represents this situation so we can eventually solve for n. Let's go through each part of the sentence and translate it into mathematical language.

Decoding the Problem

"A number, n" simply refers to our variable, n. No transformation needed here. Next, "3 times itself" means we multiply the number n by 3, giving us 3n. Then comes the phrase "15 less than 3 times itself." This indicates that we subtract 15 from 3n, resulting in the expression 3n - 15. Finally, we add the original number, n, to this expression, and the entire sum equals 101. Putting it all together, we get the equation:

3n - 15 + n = 101

This equation tells us that if we take three times the number n, subtract 15, and then add n again, we will end up with 101. This perfectly mirrors the problem's description, making it the correct equation to solve for n. Now, let's examine why the other options are incorrect.

Analyzing the Incorrect Options

To ensure we fully understand the problem, let's dissect why the other provided equations don't fit the scenario. This will reinforce our understanding of how to translate word problems into mathematical expressions accurately.

• Option B: 3n + 15 + n = 101

  This equation represents a situation where we're adding 15 to 3 times n, instead of subtracting 15. The problem clearly states "15 less than 3 times itself," indicating subtraction, not addition. Therefore, this option is incorrect. It changes the fundamental relationship described in the word problem. Instead of reducing the value of 3n by 15, it increases it, leading to a different result altogether. Imagine if n was 10; in the correct equation, we'd have 3(10) - 15 = 15, but in this incorrect equation, we'd have 3(10) + 15 = 45, a significant difference.

• Option C: 3n - 15 - n = 101

  In this equation, we are subtracting n instead of adding it. The problem states that the number n is added to "15 less than 3 times itself." This equation misinterprets the addition as subtraction, leading to an inaccurate representation of the problem. While it correctly subtracts 15 from 3n, it then incorrectly subtracts n as well. This fundamentally alters the equation and will not yield the correct solution for n. To illustrate, if n were 10, the correct operation after "15 less than 3 times itself" should be adding 10, but here, we're subtracting 10, changing the entire equation's outcome.

• Option D: 3n + 15 - n = 101

  This equation makes two critical errors. First, it adds 15 to 3n instead of subtracting, as we've discussed. Second, it subtracts n instead of adding it. This equation completely misrepresents the relationships described in the problem, making it definitively incorrect. It not only reverses the subtraction of 15 but also changes the addition of n to a subtraction. This double error compounds the inaccuracy, ensuring that this equation is nowhere near the correct representation of the original problem.

By understanding why these options are incorrect, we solidify our comprehension of translating word problems into algebraic equations.

The Correct Equation: A Deep Dive

The correct equation, 3n - 15 + n = 101, accurately captures the essence of the problem. Let's dissect it further to highlight its nuances and ensure a complete understanding. The phrase "15 less than 3 times itself" is perfectly translated into the term 3n - 15. This means we first multiply the number n by 3 and then subtract 15 from the result. The order of operations is crucial here; we must perform the multiplication before the subtraction to adhere to the problem's description. The subsequent addition of n is represented by the + n term. This signifies that we are adding the original number n to the result of 3n - 15. The problem explicitly states that the number n is added, reinforcing the correctness of this term. The equality to 101, indicated by = 101, signifies that the entire expression on the left-hand side of the equation is equal to 101. This is the final piece of information provided in the problem, stating the result of the addition. By combining these elements, we construct a complete and accurate equation that mirrors the word problem's description. The equation allows us to use algebraic techniques to isolate n and find its value.

Solving for n (Optional)

While the question only asks for the correct equation, let's go the extra mile and solve for n to demonstrate the utility of the equation. Here's how we can do it:

1. Combine like terms: In the equation 3n - 15 + n = 101, we can combine the terms 3n and n to get 4n. So the equation becomes 4n - 15 = 101.
2. Isolate the variable term: To isolate 4n, we add 15 to both sides of the equation: 4n - 15 + 15 = 101 + 15, which simplifies to 4n = 116.
3. Solve for n: To solve for n, we divide both sides of the equation by 4: 4n / 4 = 116 / 4, which simplifies to n = 29.

Therefore, the value of n that satisfies the given conditions is 29. This confirms that our equation is not only correct but also useful for finding the solution to the problem.

Conclusion

In conclusion, the correct equation to find the value of n is A. 3n - 15 + n = 101. This equation accurately translates the word problem into a mathematical expression, allowing us to solve for the unknown number n. By understanding the components of the equation and why the other options are incorrect, we gain a deeper appreciation for the process of translating word problems into algebraic equations. Remember, pay close attention to the wording of the problem, especially phrases like "less than" and ensure you're adding or subtracting the correct terms. With practice, you'll become a pro at solving these types of problems! And remember math its all about practice to get better, so keep practicing. If you have any math questions reach out for help there are many helpful videos and instructors who are ready to help.