Inverse Variation: Find V When P=4

Understanding inverse variation is super important in math and science. Guys, let's break down this problem step by step, making sure everyone gets it. We're given the equation $p = \frac{8}{V}$ and asked to find the value of $V$ when $p = 4$. It sounds tricky, but trust me, it’s totally manageable.

Decoding Inverse Variation

First, let’s quickly recap what inverse variation means. In simple terms, two variables vary inversely if their product is constant. Mathematically, this is represented as $xy = k$, where $k$ is a constant. In our case, we have $p = \frac{8}{V}$, which can be rewritten as $pV = 8$. This tells us that as $p$ increases, $V$ decreases, and vice versa, while their product remains constant at 8.

Now, back to our specific problem. We know that $p = 4$, and we need to find the corresponding value of $V$. We can plug in the value of $p$ into our equation: $4V = 8$. To solve for $V$, we simply divide both sides of the equation by 4: $V = \frac{8}{4} = 2$. So, when $p = 4$, the value of $V$ is 2. Therefore, the correct answer is C. 2.

Practical Examples of Inverse Variation

To solidify our understanding, let's explore some real-world examples of inverse variation. One classic example is the relationship between the speed and time it takes to travel a certain distance. If you increase your speed, the time it takes to cover the same distance decreases, assuming the distance remains constant. This can be expressed as $speed \times time = distance$. Another example is the number of workers and the time it takes to complete a job. If you increase the number of workers, the time required to finish the job decreases, assuming each worker contributes equally. This can be expressed as $workers \times time = work$.

Understanding inverse variation is not just about solving equations; it's about recognizing relationships in the world around us. Whether it's in physics, economics, or everyday life, inverse variation helps us make sense of how different quantities interact. Keep practicing, and you'll become a pro at spotting and solving inverse variation problems!

Step-by-Step Solution

Let's reiterate how to solve the problem to really nail it down:

1. Write down the given equation: $p = \frac{8}{V}$.
2. Substitute the given value of $p$: We know $p = 4$, so we substitute that into the equation: $4 = \frac{8}{V}$.
3. Solve for $V$: To isolate $V$, we can multiply both sides of the equation by $V$, giving us $4V = 8$. Then, divide both sides by 4 to get $V = \frac{8}{4}$.
4. Simplify: $V = 2$.

So the final answer is $V = 2$.

Why Other Options Are Incorrect

It's also helpful to understand why the other answer options are incorrect. This reinforces our understanding of the problem and helps avoid common mistakes.

• A. $\frac{1}{8}$: If $V = \frac{1}{8}$, then $p = \frac{8}{\frac{1}{8}} = 8 \times 8 = 64$, which is not equal to 4.
• B. $\frac{1}{2}$: If $V = \frac{1}{2}$, then $p = \frac{8}{\frac{1}{2}} = 8 \times 2 = 16$, which is also not equal to 4.
• D. 32: If $V = 32$, then $p = \frac{8}{32} = \frac{1}{4}$, which is again not equal to 4.

By checking these options, we can clearly see that only $V = 2$ satisfies the given equation when $p = 4$.

Common Mistakes to Avoid

When dealing with inverse variation problems, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

1. Misunderstanding the relationship: The most common mistake is not understanding that as one variable increases, the other decreases. Always remember that in inverse variation, the product of the two variables remains constant.
2. Incorrectly setting up the equation: Make sure you set up the equation correctly. In this case, it's $p = \frac{8}{V}$ or $pV = 8$. Mixing up the variables can lead to the wrong answer.
3. Algebraic errors: Be careful with your algebra. Simple mistakes in multiplying, dividing, or simplifying can lead to incorrect results. Always double-check your work.
4. Forgetting to substitute correctly: When substituting the given value, ensure you're putting it in the right place. For example, if $p = 4$, make sure you replace $p$ with 4 in the equation.
5. Not checking the answer: After finding the value of $V$, plug it back into the original equation to make sure it satisfies the condition. This helps catch any errors you might have made.

Further Practice

To really master inverse variation, it's essential to practice more problems. Here are a few practice questions you can try:

1. If $y$ varies inversely as $x$, and $y = 6$ when $x = 2$, find $y$ when $x = 3$.
2. Suppose $a$ varies inversely as $b$, and $a = 4$ when $b = 5$. Find $a$ when $b = 10$.
3. If $z$ varies inversely as $w$, and $z = 8$ when $w = \frac{1}{2}$, find $z$ when $w = 2$.

By working through these problems, you'll build confidence and improve your understanding of inverse variation. Remember to follow the steps we discussed earlier: set up the equation, substitute the given values, solve for the unknown variable, and check your answer.

Conclusion

In conclusion, solving inverse variation problems involves understanding the relationship between the variables, setting up the equation correctly, and carefully performing the algebraic steps. By avoiding common mistakes and practicing regularly, you can become proficient in solving these types of problems. Remember, when $p = \frac{8}{V}$ and $p = 4$, the value of $V$ is indeed 2. Keep practicing, and you'll ace those math tests in no time! Remember, math isn't scary, it's fun. Keep practicing and you'll get better and better!