Hey there, math enthusiasts! Ever stumbled upon a formula and felt a little lost? Don't worry, we've all been there. Today, we're diving into a classic problem that involves a simple yet powerful formula: p = kt. This formula pops up in various fields, from physics to economics, so understanding how to work with it is a valuable skill. We'll break down the steps to find the value of 't' when we know 'k' and 'p'. Let's get started and unlock this mathematical mystery together!
Decoding the Formula: p = kt
Before we jump into solving the problem, let's take a moment to understand what the formula p = kt actually means. This equation represents a direct proportion, which essentially means that 'p' is directly proportional to 't'. In simpler terms, as 't' increases, 'p' also increases proportionally, and vice versa. The constant 'k' is the magic number that determines the strength of this relationship. Think of 'k' as the constant of proportionality – it tells us how much 'p' changes for every unit change in 't'.
For example, if 'k' is a large number, it means that even a small change in 't' will result in a significant change in 'p'. On the other hand, if 'k' is a small number, 'p' will change more gradually with changes in 't'. This concept of direct proportionality is fundamental in many real-world scenarios. Imagine calculating the cost of buying multiple items, where 'p' is the total cost, 't' is the number of items, and 'k' is the price per item. Or think about the distance traveled at a constant speed, where 'p' is the distance, 't' is the time, and 'k' is the speed. Grasping the meaning of p = kt lays the groundwork for tackling a wide range of problems.
Understanding direct proportionality helps us in many practical situations. Let's say you're baking cookies and the recipe says you need to double all the ingredients if you want to make twice as many cookies. That's direct proportionality in action! The amount of each ingredient is directly proportional to the number of cookies you want to bake. Or, consider the relationship between the hours you work and the money you earn, assuming you have an hourly wage. The more hours you work, the more money you earn, and your hourly wage acts as the constant of proportionality. So, the next time you see p = kt or a similar equation, remember it's not just a bunch of letters – it's a powerful tool for describing relationships in the world around us.
The Challenge: Finding 't' When k = 36 and p = 144
Now that we've got a handle on the formula p = kt, let's tackle the specific problem at hand. We're given that k = 36 and p = 144, and our mission is to find the value of 't'. This is where our algebraic skills come into play. Remember, the goal is to isolate 't' on one side of the equation. To do this, we'll use the power of inverse operations. Since 't' is being multiplied by 'k', which is 36, we need to do the opposite operation – divide both sides of the equation by 36. This will effectively cancel out the 36 on the side with 't', leaving us with 't' all by itself. It's like unwrapping a present, where each step reveals a little more of the solution.
Let's walk through the process step-by-step. We start with the equation p = kt, and we know that p = 144 and k = 36. So, we can substitute these values into the equation, giving us 144 = 36t. Now, we divide both sides of the equation by 36. This gives us 144 / 36 = 36t / 36. The 36 on the right side cancels out, leaving us with 144 / 36 = t. Finally, we perform the division: 144 divided by 36 equals 4. So, we've found that t = 4! See, it's not as daunting as it might have seemed at first. By understanding the formula and using basic algebraic principles, we were able to crack the code and find the value of 't'.
Think of this process like solving a puzzle. Each piece of information we have – the formula, the values of 'k' and 'p' – is like a puzzle piece. Our goal is to fit these pieces together in the right way to reveal the solution. Dividing both sides of the equation is like making a strategic move in the puzzle, bringing us closer to the final picture. And just like with puzzles, practice makes perfect! The more you work with equations like this, the more comfortable and confident you'll become in your problem-solving abilities.
The Solution: t = 4
Alright, guys, let's recap our journey to find 't'. We started with the formula p = kt, plugged in the values k = 36 and p = 144, and then used our algebra skills to isolate 't'. Remember, we divided both sides of the equation by 36 to get 't' by itself. After performing the division, we arrived at the answer: t = 4. This means that when 'k' is 36 and 'p' is 144, the value of 't' is precisely 4. So, the correct answer from the options provided is B) 4. Great job to everyone who followed along and worked through the problem with us!
But the fun doesn't stop here! Understanding how to solve for 't' in this formula opens the door to solving for other variables as well. What if we were given 'p' and 't' and asked to find 'k'? The process would be similar – we'd still use inverse operations to isolate the variable we're looking for. This flexibility is one of the beautiful things about algebra. Once you grasp the fundamental principles, you can apply them to a wide variety of problems. So, keep practicing, keep exploring, and keep unlocking those mathematical mysteries!
And remember, even if you encountered some bumps along the way, that's perfectly okay. Math is a journey, and every mistake is an opportunity to learn and grow. The important thing is to keep trying, keep asking questions, and keep building your understanding. With perseverance and a positive attitude, you can conquer any mathematical challenge that comes your way. So, let's celebrate our success in finding 't' and look forward to the next adventure in the world of math!
Why is This Important? Real-World Applications
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