Rounding To 300: Find The Digit Sum!

Hey guys! Let's dive into a fun math problem that involves rounding numbers. We're going to figure out the largest and smallest numbers that round to 300 when rounded to the nearest hundred. And then, we'll add up all their digits. Sounds like a plan? Let's get started!

Understanding Rounding to the Nearest Hundred

Before we jump into finding the specific numbers, let's quickly refresh our understanding of rounding to the nearest hundred. When we round a number to the nearest hundred, we're essentially finding the closest multiple of 100. Think of it like this: if a number is closer to 300 than it is to 200 or 400, we round it to 300.

The general rule is: if the tens digit is 5 or greater, we round up to the next hundred. If the tens digit is 4 or less, we round down to the current hundred. For example, 250, 260, 270, 280, and 290 all round up to 300. On the other hand, 310, 320, 330, and 340 all round down to 300. This is the golden rule of rounding, and it's super important to keep in mind as we solve this problem.

Why is understanding this important? Well, it helps us define the boundaries. The smallest number that rounds to 300 will be the lowest possible number that still rounds up to 300. Similarly, the largest number that rounds to 300 will be the highest possible number that still rounds down to 300. These boundaries are crucial because they give us the precise numbers we need to work with. Without knowing the rules, we would just be guessing, and nobody wants to guess in math! Plus, having a solid grasp on rounding helps with estimating, which is a handy skill in everyday life, like when you're trying to figure out if you have enough money to buy everything in your shopping cart. So, rounding isn't just a math concept; it's a practical tool.

Finding the Largest Number That Rounds to 300

Okay, so what's the largest number that rounds to 300? To find this, we need to think about the highest number that is still closer to 300 than it is to 400. Remember, if we hit 350, we're rounding up to 400. So, the largest number must be just below 350. Therefore, the largest number that rounds to 300 is 349. This is the upper limit, the highest we can go without tipping over into the next hundred.

Why 349? Because if we add just 1 to it, we get 350, and 350 rounds up to 400. It's like being on the edge of a cliff – one more step, and you fall off! So, 349 is our safe spot, the largest number that still adheres to our rounding rule. It's super important to understand this because sometimes the wording of the problem can be tricky, and you might be tempted to go higher. But sticking to the rule ensures we get the right answer.

Now, let's break down why this works. The tens digit of 349 is 4, which is less than 5. That's why it rounds down (or in this case, stays at) 300. If the tens digit were 5 or more, it would round up to 400. So, by keeping the tens digit at 4, we guarantee that it rounds to 300. And since we want the largest number, we make the ones digit as big as possible, which is 9. Hence, 349 is our winner!

Finding the Smallest Number That Rounds to 300

Now, let's switch gears and find the smallest number that rounds to 300. To do this, we need to think about the lowest number that is still closer to 300 than it is to 200. This is the lower limit. What's the smallest number that rounds up to 300? It's 250! Anything below 250 would round down to 200. Therefore, 250 is the smallest number that rounds to 300. This is the lower limit, the lowest we can go without falling back into the previous hundred.

Why 250? Because the tens digit is 5, which means it rounds up to 300. If it were any smaller, like 249, it would round down to 200. So, 250 is our critical point, the smallest number that meets the criteria. It's like setting a minimum height requirement for a ride – you have to be at least this tall to get on! Similarly, a number has to be at least 250 to round to 300.

To understand this better, consider the numbers around 250. If we have 249, it rounds down to 200. If we have 251, it rounds up to 300. So, 250 is the boundary, the exact point where the rounding changes. This is why it's crucial to consider the numbers on either side of our target. It helps us confirm that we've indeed found the smallest number that rounds to 300. Remember, precision is key in math!

Calculating the Sum of the Digits

Alright, we've found our two key numbers: 349 (the largest) and 250 (the smallest). Now comes the fun part: adding up their digits!

For 349, the digits are 3, 4, and 9. So, the sum of the digits is 3 + 4 + 9 = 16.

For 250, the digits are 2, 5, and 0. So, the sum of the digits is 2 + 5 + 0 = 7.

Now, we add these two sums together: 16 + 7 = 23.

So, the sum of the digits of the largest number that rounds to 300 and the smallest number that rounds to 300 is 23. And that's our final answer! It's like solving a puzzle where each piece fits perfectly together to reveal the solution. In this case, our pieces were the rounding rules, the largest and smallest numbers, and the digit sums. And when we put them all together, we got 23. High five!

Conclusion

Great job, everyone! We successfully found the largest and smallest numbers that round to 300 and then calculated the sum of their digits. It's a great exercise in understanding rounding rules and paying attention to detail. Remember, math isn't just about getting the right answer; it's about understanding why the answer is right. So, keep practicing, keep exploring, and most importantly, keep having fun with math! You guys rock!