Solving Math Problems: Finding The Value Of Ψ

Hey guys! Let's dive into a cool math problem. We're going to figure out the value of a certain expression involving a special symbol, Ψ (Psi). Don't worry, it's not as scary as it looks! We'll break it down step-by-step, making it super easy to follow. This is a great example of how you can use mathematical operations and algebraic thinking to solve problems. So, grab your pencils and let's get started!

Understanding the Problem and the Ψ Operation

Alright, so the problem asks us to find the value of (-3/2 Ψ 1/6) : (5 Ψ -2). What's that Ψ thing all about? Well, the problem tells us how to use it: α Ψ β = α + 2β - αβ. This is the key to solving the problem. Think of Ψ as a custom-made operation, like addition or multiplication, but with its own special rules. The formula α Ψ β = α + 2β - αβ tells us exactly what to do with two numbers, α (alpha) and β (beta), when they're connected by Ψ. It's like a recipe! For instance, if α = 2 and β = 3, then 2 Ψ 3 = 2 + 2(3) - 2(3) = 2 + 6 - 6 = 2. See? Not so bad, right?

This type of problem often shows up in math contests and exams. It tests your ability to understand new operations and apply them correctly. The biggest challenge is usually just staying organized and making sure you substitute the numbers into the formula correctly. We'll be careful and go slow, so you can totally ace this! Remember, the most important thing is to understand what the question is asking and how the given formula works. The rest is just plugging in the numbers and doing the arithmetic. Keep in mind that we're dealing with fractions here, but don't let that throw you. With a little care, you can handle them just fine. Remember to follow the order of operations (PEMDAS/BODMAS) to avoid any mistakes. In this case, we have a division problem so, first, we'll solve both sides of the division, and then, we'll do the division itself. Let's do this!

To make things easier, let's look at the formula again. α Ψ β = α + 2β - αβ. You'll use this formula twice: once for the part in the first set of parentheses (-3/2 Ψ 1/6) and once for the part in the second set of parentheses (5 Ψ -2). Then, finally, we'll divide the result of the first part by the result of the second part. Got it? Perfect!

Solving (-3/2 Ψ 1/6)

Okay, let's tackle the first part: (-3/2 Ψ 1/6). Here, α = -3/2 and β = 1/6. Now, we just plug these values into our formula: α Ψ β = α + 2β - αβ. So, we get: -3/2 Ψ 1/6 = -3/2 + 2(1/6) - (-3/2)(1/6). Let's simplify this step by step. First, 2(1/6) = 2/6 = 1/3. Next, -(-3/2)(1/6) = 3/12 = 1/4. So our expression becomes: -3/2 + 1/3 + 1/4. We need to add these fractions. To do that, we need a common denominator. The smallest common denominator for 2, 3, and 4 is 12. Let's rewrite each fraction with a denominator of 12:

• -3/2 = -18/12
• 1/3 = 4/12
• 1/4 = 3/12

Now, let's add them: -18/12 + 4/12 + 3/12 = (-18 + 4 + 3)/12 = -11/12. So, (-3/2 Ψ 1/6) = -11/12. See? Not too tough, right? We've successfully calculated the value of the first part of our original problem. It's all about carefully substituting the numbers and then doing the arithmetic correctly. Make sure to double-check your calculations at each step to avoid errors. The most common mistakes come from adding and subtracting fractions, so always take your time and find the common denominator correctly.

Great job on solving the first part. Now we're halfway there. We're going to use the same process for the second part of the problem. Remember, practice makes perfect. Keep going! Keep up the great work!

Solving (5 Ψ -2)

Now, let's move on to the second part: (5 Ψ -2). This time, α = 5 and β = -2. Using the same formula, α Ψ β = α + 2β - αβ, we plug in our values: 5 Ψ -2 = 5 + 2(-2) - 5(-2). Let's simplify this: 2(-2) = -4 and -5(-2) = 10. So our expression becomes: 5 - 4 + 10. Now we can just do the addition and subtraction from left to right: 5 - 4 = 1, and then 1 + 10 = 11. Therefore, (5 Ψ -2) = 11. That was quick, wasn't it? See how the same formula can be used in different scenarios? Again, make sure to follow the order of operations: multiplication and division before addition and subtraction. Double-check all calculations to minimize errors. Also, be careful with negative signs when multiplying and subtracting. When working with mathematical problems, you have to be detail-oriented, which means paying attention to all the signs and operations involved in the equations. Remember, even if the concepts may seem complex at first, with practice and attention, these calculations become second nature. You'll become a master of these operations in no time!

We are in the last step. Now, let's divide the result of the first part by the result of the second part. Great job, guys!

Calculating the Final Result

Okay, we're at the finish line! Remember, the original problem was (-3/2 Ψ 1/6) : (5 Ψ -2). We've found that (-3/2 Ψ 1/6) = -11/12 and (5 Ψ -2) = 11. So now we just need to do the division: -11/12 : 11. This is the same as -11/12 / 11. To divide by a number, we can multiply by its reciprocal. The reciprocal of 11 is 1/11. So we have -11/12 * 1/11. Multiply the numerators and the denominators: (-11 * 1) / (12 * 11) = -11/132. We can simplify this fraction by dividing both the numerator and the denominator by 11: -11/132 = -1/12. Therefore, the answer to our problem is -1/12!

This problem showed us the importance of understanding the question, the formula, and how to apply it correctly. We also practiced our fraction arithmetic. With a little practice, these kinds of problems become much easier. Remember to keep practicing and, most importantly, have fun with math!

Conclusion and Answer Choice

We have successfully navigated through the math problem, and we arrived at an answer. Let's summarize our findings:

1. We understood the Ψ (Psi) operation formula.
2. We calculated (-3/2 Ψ 1/6) to be -11/12.
3. We calculated (5 Ψ -2) to be 11.
4. We divided -11/12 by 11 to get -1/12.

Therefore, the correct answer is a) -1/12. We hope this explanation helped you! Keep up the awesome work!