French Club Bake Sale How Many Pastries To Sell

Hey everyone! Let's dive into a tasty math problem today. Our French club is hosting a bake sale, and their goal is to raise at least $135. They're selling each pastry for $2.50, and we need to figure out how many pastries they need to sell to meet their goal. It’s a classic scenario where math meets real-life fundraising, and we’re going to break it down step by step.

Setting Up the Inequality

So, how do we translate this word problem into a mathematical inequality? The key here is to identify the unknowns and the constraints. Let’s use 'p' to represent the number of pastries they need to sell. Since each pastry costs $2.50, the total amount they'll earn from selling 'p' pastries is $2.50 multiplied by 'p', which we write as $2.50p$. Now, they want to raise at least $135. This “at least” is super important because it tells us we're dealing with an inequality, not just a simple equation. When we say “at least,” it means the amount they raise should be greater than or equal to $135. So, we can write our inequality as:

$2. 50p \geq 135$

This inequality is the heart of our problem. It tells us that the total revenue from selling pastries ($2.50p$) must be greater than or equal to their fundraising goal of $135. Think of it like this: if they sell enough pastries, they'll hit their target, and anything more is just a bonus! This simple inequality is a powerful tool that lets us solve for the minimum number of pastries they need to sell. Guys, understanding how to set up these inequalities is crucial because it’s the first step in solving many real-world problems, not just bake sales!

Solving the Inequality

Now that we've set up our inequality, $2.50p \geq 135$, it's time to solve it. Solving inequalities is very similar to solving equations, but there’s one key difference we'll talk about later. Our goal here is to isolate 'p' on one side of the inequality to figure out the minimum number of pastries. To do this, we need to get rid of the $2.50 that's multiplying 'p'. The way we do that is by dividing both sides of the inequality by $2.50$. Remember, what we do to one side, we must do to the other to keep the inequality balanced. So, let’s do it:

$\frac{2.50p}{2.50} \geq \frac{135}{2.50}$

On the left side, the $2.50 in the numerator and denominator cancel each other out, leaving us with just 'p'. On the right side, we need to divide 135 by 2.50. If you do the math (grab a calculator if you need to!), you’ll find that 135 divided by 2.50 is 54. So, our inequality now looks like this:

$p \geq 54$

This is our solution! It tells us that 'p', the number of pastries they need to sell, must be greater than or equal to 54. In other words, they need to sell at least 54 pastries to reach their goal of $135. Isn't it cool how we've used math to figure out a real-world scenario? Now, about that key difference between solving inequalities and equations... In most cases, you solve inequalities just like equations. However, there's a very important rule to remember: if you multiply or divide both sides of an inequality by a negative number, you have to flip the direction of the inequality sign. This is crucial to keep in mind, but luckily, in our case, we divided by a positive number ($2.50), so we didn't need to worry about it. Always double-check for this when you're solving inequalities, guys!

Interpreting the Solution

Okay, so we've crunched the numbers and found that $p \geq 54$. But what does this really mean for the French club's bake sale? Interpreting the solution in the context of the problem is super important. It's not just about getting a number; it's about understanding what that number tells us in the real world. Our inequality tells us that the number of pastries ('p') the French club needs to sell must be greater than or equal to 54. This means they need to sell a minimum of 54 pastries to reach their goal of raising $135. If they sell exactly 54 pastries, they'll make exactly $135 (54 pastries * $2.50/pastry = $135). But what if they want to raise more money? Well, they can sell more pastries! The inequality $p \geq 54$ tells us that any number of pastries greater than 54 will also meet their goal. For example, if they sell 60 pastries, they'll make $150 (60 pastries * $2.50/pastry = $150), which is more than their goal. So, the solution isn't just a single number; it's a range of numbers. In this case, it’s 54 and any number greater than 54. This is a key difference between inequalities and equations. Equations usually have one specific solution, while inequalities have a range of solutions. Guys, understanding how to interpret solutions in real-world contexts is a crucial skill. It's not enough to just solve the math; you need to understand what the answer means in the situation you're dealing with. In the case of the bake sale, the French club now knows the minimum number of pastries they need to bake, but they also have the flexibility to bake more if they want to aim for a higher fundraising target.

Checking the Solution

Alright, we've set up the inequality, solved it, and interpreted the solution. But before we declare victory, there's one more crucial step: checking our solution. Checking our work is super important because it helps us catch any mistakes and makes sure our answer makes sense in the context of the problem. So, how do we check our solution to the inequality $p \geq 54$? The easiest way is to plug a number that satisfies the inequality back into the original inequality and see if it holds true. Let's pick 54, the minimum number of pastries. If we plug 54 into our original inequality, $2.50p \geq 135$, we get:

$2. 50 * 54 \geq 135$

Now, let’s do the math: $2.50 multiplied by 54 is exactly $135. So, our inequality becomes:

$135 \geq 135$

This is true! $135 is indeed greater than or equal to $135. This confirms that 54 is a valid solution. But remember, inequalities have a range of solutions. So, let's try a number greater than 54, say 60. If we plug 60 into our original inequality, we get:

$2. 50 * 60 \geq 135$

$2.50 multiplied by 60 is $150, so our inequality becomes:

$150 \geq 135$

This is also true! $150 is greater than $135. This further confirms that our solution makes sense. We can also try a number less than 54 to see what happens. Let's try 50:

$2. 50 * 50 \geq 135$

$2.50 multiplied by 50 is $125, so our inequality becomes:

$125 \geq 135$

This is not true! $125 is less than $135. This tells us that selling 50 pastries is not enough to meet their goal. Guys, checking your solution with different values is a great way to make sure you've solved the inequality correctly and that your answer makes sense in the real world. It's like a final safety net before you finalize your answer.

Conclusion

So, there you have it! We've successfully helped the French club figure out how many pastries they need to sell to reach their fundraising goal. We started with a word problem, translated it into a mathematical inequality ($2.50p \geq 135$), solved the inequality ($p \geq 54$), interpreted the solution (they need to sell at least 54 pastries), and even checked our work to make sure we got it right. This whole process highlights the power of math in solving real-world problems. It's not just about abstract numbers and symbols; it's about using those tools to make informed decisions and achieve goals. Whether it's planning a bake sale, budgeting for a project, or figuring out discounts, inequalities are a valuable tool in our mathematical toolkit. Guys, remember that practice makes perfect. The more you work with inequalities and other mathematical concepts, the more comfortable and confident you'll become in applying them to real-life situations. So, keep practicing, keep asking questions, and keep exploring the amazing world of math! And who knows, maybe you'll be inspired to organize your own bake sale or fundraising event, armed with the mathematical knowledge to make it a success!