Key Concepts: the Principle of “Same Sum, Smaller Difference, Bigger Product”

Hello young mathematicians! Today, we’re going to solve an interesting problem together. Imagine you’re selling a toy, and you want to find the best price to sell it at to make the most money. Sounds fun, right? Let’s dive in!

Here’s the situation: You have a toy that you can sell for 15. When you do, you can sell 500 of them. But, if you increase the price by 1, you sell 20 fewer toys. The question is: What price should you set to earn the most money?

Let’s call the price you set $x$. According to the problem:

Total sales = Price * Quantity = $x * (500 - (x-15)*20)$ = $x * (500 - 20x + 300)$ = $x * (800 - 20x)$ = $20 * x * (40-x)$

Now, there’s a cool rule in mathematics that says: If the sum of two numbers is constant, their product is maximum when the numbers are equal. Here, our two numbers are $x$ and $40-x$, and their sum is always 40. So, to maximize the product (which is our total sales), we should set $x = 40-x$.

Solving this equation gives us $2x = 40$, so $x=20$.

So, the best price to sell your toy at is 20. At this price, your total sales will be $20 * x * (40-x) = 20 *20 * 20 = 8000$ dollars. That’s a lot of money from selling toys!

I hope you enjoyed this journey into the world of math and business. Remember, math is not just about numbers, it’s also about finding the best solutions to real-world problems. So keep practicing, and you’ll be a master problem solver in no time!