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

Hello young mathematicians! Today, we are going on an exciting journey to solve a fun math problem. This problem comes from the world of Elementary Olympiad Math. Don’t worry, it’s not as scary as it sounds. In fact, it’s quite fun!

Our problem for today is: If $2a + 3b = 84$, what is the maximum value of $ab$?

At first glance, this might seem like a tricky problem. But don’t worry, we have a secret weapon in our math toolbox: the concept of “When the sum is fixed, the product is maximized when the numbers are equal”.

Let’s break it down. The sum of $2a$ and $3b$ is fixed at 84. But the sum of $a$ and $b$ is not fixed. So, we can make $2a$ equal to $3b$ to maximize the product $ab$.

So, let’s make $2a = 3b$. Now, we can substitute this into our original equation $2a + 3b = 84$.

This gives us $3b + 3b = 84$. Simplifying this, we find that $b = 14$.

Now, we can find $a$ by using our equality $2a = 3b$. Substituting $b = 14$ into this, we get $a = 21$.

Finally, we can find the maximum value of $ab$ by multiplying $a$ and $b$ together. This gives us $ab = 21 * 14 = 294$.

And there you have it! The maximum value of $ab$ is 294.

I hope you enjoyed this fun journey into Elementary Olympiad Math. Remember, math is not just about finding the right answer, but also about understanding the process. So keep practicing, keep exploring, and keep having fun with math!