Solve For Package Price: $5(p-2.50)=3p$

Hey guys, let's dive into a cool math problem today that's all about figuring out the price of things. We've got this scenario where some packages are on sale, and some aren't, and we need to find out the original price of the ones that aren't on sale. The problem gives us a neat equation: $5(p-2.50)=3p$. This equation is super helpful because it directly relates the prices of the sale packages to the prices of the non-sale packages. Here, $p$ represents the original price of a package that is not on sale. Our mission, should we choose to accept it, is to solve for $p$ and uncover that hidden price. This isn't just about crunching numbers; it's about understanding how algebraic equations can model real-world situations, like discounts and pricing strategies. So, grab your calculators, get comfy, and let's break down this equation step-by-step to find the answer. We'll explore the logic behind the equation and how manipulating it leads us to the solution. Get ready to flex those math muscles!

Understanding the Equation: $5(p-2.50)=3p$

Alright team, let's really unpack this equation: $5(p-2.50)=3p$. It might look a little intimidating at first, but trust me, it tells a story. The left side of the equation, $5(p-2.50)$, represents the total cost of five packages that are on sale. Why $p-2.50$? Because $p$ is the original price (the price of a package not on sale), and the sale price is $2.50 less than that original price. So, $(p-2.50)$ is the price of one sale package. Multiply that by 5, and you get the total cost for five sale packages. Now, the right side of the equation, $3p$, represents the total cost of three packages that are not on sale. Since $p$ is already defined as the price of a non-sale package, $3p$ is simply the cost of three of those. The equals sign, '=', is the key here. It tells us that the total cost of the five sale packages is exactly the same as the total cost of the three non-sale packages. This is the core relationship the problem is highlighting. Think about it: if you buy five items at a discount, and the total amount you spend is the same as buying three of the full-priced items, that gives us a powerful way to determine the original price. This setup is super common in retail scenarios where promotions are designed to make a certain number of discounted items equivalent in cost to a smaller number of full-priced items. It's all about finding that sweet spot where the value proposition makes sense. We're essentially equating two different purchasing scenarios that result in the same expenditure. This is the beauty of algebra – it distills complex comparisons into simple, solvable equations. So, before we even start solving, just appreciating what each part of the equation means is a huge step. It translates the word problem into a clear mathematical statement.

Step-by-Step Solution to Find the Package Price

Now that we've got a solid grasp on what our equation $5(p-2.50)=3p$ means, let's roll up our sleeves and solve for $p$. This is where the magic happens, guys! Our goal is to isolate $p$ on one side of the equation.

Step 1: Distribute on the left side.

The first thing we need to do is get rid of those parentheses on the left side. We do this by distributing the 5 to both terms inside the parentheses: $p$ and $-2.50$.

So, $5 \times p = 5p$ and $5 \times -2.50 = -12.50$.

Our equation now looks like this: $5p - 12.50 = 3p$.

Step 2: Get all the $p$ terms on one side.

We want to gather all our $p$ terms together. It's usually easier to move the smaller $p$ term. In this case, $3p$ is smaller than $5p$. So, we'll subtract $3p$ from both sides of the equation to keep it balanced.

$5p - 12.50 - 3p = 3p - 3p$

This simplifies to: $2p - 12.50 = 0$.

Step 3: Isolate the $p$ term.

Now we have the $p$ terms on one side and a constant on the other. To get the $2p$ term by itself, we need to move the $-12.50$ to the other side. We do this by adding $12.50$ to both sides of the equation.

$2p - 12.50 + 12.50 = 0 + 12.50$

This gives us: $2p = 12.50$.

Step 4: Solve for $p$.

We're almost there! We have $2p$ equals $12.50$. To find the value of a single $p$, we need to divide both sides of the equation by 2.

$\frac{2p}{2} = \frac{12.50}{2}$

$p = 6.25$.

And there you have it! The price of a package that is not on sale, $p$, is $6.25. It's pretty neat how following a few algebraic steps can lead us straight to the answer, right? We took a word problem, translated it into an equation, and then systematically solved it. This is the power of mathematics in action!

Verifying the Solution: Does $p = 6.25$ Work?

So, we found that $p = 6.25$. But is this answer correct? The best way to be sure is to plug this value back into our original equation, $5(p-2.50)=3p$, and see if both sides are equal. This is called verification, and it's a super important step to make sure we haven't made any slip-ups along the way. Let's do it together!

First, let's figure out the price of a sale package. If the non-sale price ($p$) is $6.25$, then the sale price is $p - 2.50$.

Sale Price = $6.25 - 2.50 = 3.75$.

Now, let's calculate the total cost for five sale packages:

Cost of 5 sale packages = $5 \times (p - 2.50) = 5 \times 3.75$.

$5 \times 3.75 = 18.75$.

So, the left side of our original equation, $5(p-2.50)$, equals $18.75.

Next, let's calculate the total cost for three non-sale packages:

Cost of 3 non-sale packages = $3 \times p = 3 \times 6.25$.

$3 \times 6.25 = 18.75$.

So, the right side of our original equation, $3p$, also equals $18.75.

Look at that! $18.75 = 18.75$. Since both sides of the equation are equal when $p = 6.25$, our solution is correct! This verification step confirms that our calculation was spot on. It shows that five packages bought at $3.75 each (which is $2.50 off the original price of $6.25) cost the same amount as three packages bought at the full price of $6.25 each. This kind of checking is crucial in math and in life – always double-check your work to ensure accuracy. It builds confidence and solidifies understanding. Pretty cool, huh?

The Real-World Application of This Math Problem

Okay guys, so we just solved a math problem and found that $p = 6.25$. But what does this actually mean in the real world? This kind of scenario pops up more often than you might think! Imagine a store running a promotion: "Buy 5 items on sale, and your total cost is the same as buying 3 of them at full price!" This problem is a perfect example of how businesses use pricing strategies to encourage customers to buy more. They might offer a discount, but by selling you five items instead of three, they can still achieve a certain revenue target or clear inventory.

Let's break down the implication: The non-sale price of a package is $6.25. The sale price is $6.25 - 2.50 = $3.75. The store is essentially saying, "If you buy five of these items at $3.75 each, you'll spend a total of $18.75. That's the same amount you'd spend if you bought just three of them at the regular price of $6.25 each." This is a clever way to move more product. For the customer, it means that if they need more than three of these items, buying five on sale is a good deal because they are essentially getting the extra two items for free (relative to the cost of three non-sale items). If they only needed, say, two items, buying them at the sale price of $3.75 might still be cheaper than the full price of $6.25$, but the real value of the promotion is unlocked when you reach that threshold where the total cost matches the purchase of fewer full-priced items.

This problem also highlights the importance of understanding value and cost. By solving this, you're not just doing homework; you're learning to analyze deals, compare prices, and make smarter purchasing decisions. Whether you're a consumer trying to get the best bang for your buck or a business owner setting prices, these mathematical principles are fundamental. It's about understanding the relationship between quantity, price, and discount. So, next time you see a "buy X get Y at a discount" deal, you can bet there's some algebra at play behind the scenes, making sure it works out favorably for the seller, while potentially offering a great deal for the savvy shopper. It’s all about finding that equilibrium where both parties feel like they're getting a good outcome. This math is practical, folks!

Conclusion: The Price is Right!

So, there you have it, everyone! We started with a word problem that described a pricing scenario and were given an equation: $5(p-2.50)=3p$. Through a clear, step-by-step process involving distribution, rearranging terms, and isolating the variable, we successfully solved for $p$. We found that the price of a package that is not on sale, $p$, is $6.25. We then took it a step further and verified our answer by plugging $p = 6.25$ back into the original equation. This verification showed that both sides equaled $18.75$, confirming our solution is accurate. The real-world implication is fascinating: five sale packages (priced at $3.75 each) cost the exact same as three non-sale packages (priced at $6.25 each). This illustrates how discounts can be structured to encourage larger purchases while maintaining revenue. Math, guys, is not just abstract concepts; it’s a powerful tool for understanding and navigating the world around us, from store shelves to financial planning. Keep practicing, keep questioning, and you'll find that solving these problems becomes second nature. Great job today, team!