Solve P/3 = 12: Best Methods

Let's break down the best ways to solve the equation $\frac{p}{3}=12$. Understanding how to isolate the variable 'p' is key to finding the solution. We'll explore each option to determine which methods are valid and why.

Understanding the Equation

Before diving into the options, it's crucial to understand what the equation $\frac{p}{3}=12$ means. It states that some number, represented by 'p', when divided by 3, equals 12. Our goal is to find the value of 'p' that makes this statement true. To do this, we need to isolate 'p' on one side of the equation. This involves using inverse operations. Remember that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the balance and equality. This principle is fundamental to solving algebraic equations. By carefully applying inverse operations, we can systematically eliminate the numbers and operations surrounding the variable until we have 'p' all by itself, revealing its value. The beauty of algebra lies in its ability to represent unknown quantities and manipulate equations to uncover those unknowns. Understanding the basic principles and applying them consistently will empower you to solve a wide range of equations with confidence. Furthermore, always double-check your answer by substituting the value you find back into the original equation to ensure that it holds true. This is a simple yet effective way to verify your solution and catch any potential errors. In this specific case, we are aiming to find a value for 'p' that, when divided by 3, gives us 12. This requires us to think about the inverse operation of division, which is multiplication. By multiplying both sides of the equation by 3, we will effectively undo the division and isolate 'p'. This approach aligns with the core principles of algebraic manipulation and will lead us to the correct solution.

Evaluating the Options

Let's examine each option provided and determine its validity in solving the equation.

A. Subtracting 3 from both sides of the equation

Subtracting 3 from both sides would change the equation to $\frac{p}{3} - 3 = 12 - 3$, which simplifies to $\frac{p}{3} - 3 = 9$. This doesn't help isolate 'p'. The original equation involves 'p' being divided by 3; subtracting 3 introduces a new term and doesn't undo the division. Think of it like trying to untie a knot by pulling on a different strand – it simply won't work. To effectively isolate 'p', we need to address the division directly. Subtracting a constant from both sides might be useful in other types of equations, but in this specific scenario, it's not the correct approach. The key is to identify the operation that is currently affecting 'p' and then apply the inverse operation to undo it. In this case, since 'p' is being divided by 3, the inverse operation is multiplication by 3. Subtracting 3 simply adds another layer of complexity to the equation without moving us closer to the solution. Therefore, this option is incorrect. Understanding the relationship between operations and their inverses is crucial for solving algebraic equations efficiently and accurately. By recognizing that division and multiplication are inverse operations, we can quickly identify the appropriate step to isolate the variable and find its value. Furthermore, it's important to remember that the goal is to simplify the equation, not to complicate it further. Subtracting 3 from both sides in this case would only make the equation more complex and would not lead us to the correct answer. Therefore, we can confidently rule out this option as a valid method for solving the equation.

B. Multiplying both sides of the equation by 3

Multiplying both sides by 3 gives us $3 * \frac{p}{3} = 3 * 12$. This simplifies to $p = 36$. This correctly isolates 'p' and solves the equation. Multiplying by 3 cancels out the division by 3, leaving 'p' alone on one side of the equation. This is the direct and most efficient way to solve for 'p'. To illustrate further, consider the equation as a balanced scale. The left side, $\frac{p}{3}$, is equal in weight to the right side, 12. When we multiply both sides by 3, we are essentially increasing the weight on both sides by the same factor, maintaining the balance. This allows us to isolate 'p' and determine its value. Therefore, multiplying both sides of the equation by 3 is a valid and effective method for solving for 'p'. This approach aligns with the fundamental principles of algebraic manipulation and demonstrates a clear understanding of inverse operations. It's also worth noting that this method is applicable to a wide range of similar equations where a variable is being divided by a constant. By consistently applying this technique, you can confidently solve for the variable and find its value.

C. Dividing both sides of the equation by 3

Dividing both sides by 3 would result in $\frac{p}{3} / 3 = 12 / 3$, which simplifies to $\frac{p}{9} = 4$. This does not isolate 'p'. It actually makes the situation worse by further dividing 'p'. Remember, we want to undo the division, not reinforce it. Dividing both sides by 3 only complicates the equation and moves us further away from the solution. It's crucial to understand that division is the inverse of multiplication, and vice versa. In this case, since 'p' is already being divided by 3, we need to multiply by 3 to isolate it. Dividing again would be counterproductive and would not lead us to the correct answer. Therefore, this option is incorrect. Think of it as trying to put something together by taking it apart further. It simply doesn't make sense. The goal is to simplify the equation and isolate the variable, and dividing both sides by 3 would only make it more complex. Furthermore, it's important to remember that the operations we perform on both sides of the equation must be done with the intention of isolating the variable. Dividing both sides by 3 in this scenario does not achieve that goal and therefore is not a valid method for solving the equation.

D. Substituting 4 for p

Substituting 4 for 'p' gives us $\frac{4}{3} = 12$, which is false. This is a method for checking a potential solution, not for solving the equation in the first place. While substitution is a valuable tool for verifying your answer, it's not a method for finding the solution initially. Substitution involves plugging in a value for the variable and seeing if the equation holds true. In this case, substituting 4 for 'p' results in a false statement, indicating that 4 is not the solution to the equation. However, this doesn't mean that substitution is not useful. On the contrary, it's an essential step in the problem-solving process. After you have found a potential solution, you should always substitute it back into the original equation to ensure that it satisfies the equation. If it does, then you can be confident that you have found the correct answer. If it doesn't, then you know that you need to go back and check your work. Therefore, while substitution is not a method for solving the equation in the first place, it is a crucial tool for verifying your answer and ensuring its accuracy. In summary, substituting 4 for 'p' is a way to check if 4 is a solution, but it doesn't tell us how to find the solution. This option is therefore not a method to solve the equation.

Conclusion

The correct method to solve the equation $\frac{p}{3}=12$ is B. multiplying both sides of the equation by 3. This isolates 'p' and reveals its value to be 36. Option D, substituting 4 for p, is only a method to test a potential solution, not to solve for p. Remember to always use inverse operations to isolate the variable you are trying to solve for.

For more information on solving algebraic equations, you can visit Khan Academy's Algebra Basics.