Calculating The Sum Of 5 New Numbers To Increase The Average

Hey guys! Let's dive into a cool math problem that involves averages and sums. We've got a set of 15 numbers, and their average is 130. Now, we're adding 5 new numbers, and this addition bumps up the average by a whopping 60. The challenge? Figuring out the sum of these 5 new numbers. Sounds like a fun puzzle, right? Let's break it down step by step and see how we can crack this.

Understanding the Initial Situation

Let's start with understanding the initial situation. We know we have 15 numbers, and their average is 130. What does this tell us? Well, the average is the sum of all the numbers divided by the count of the numbers. So, to find the total sum of these 15 numbers, we simply multiply the average by the count. That's 130 multiplied by 15. Calculating this, we get 1950. This means the sum of our original 15 numbers is 1950. This is a crucial piece of information because it gives us a baseline to work with. We know where we started, and now we can see how adding new numbers changes things. It’s like setting the stage for the rest of our calculations. Without this initial sum, figuring out the impact of the new numbers would be much harder. So, remember, the sum of the original 15 numbers is 1950. Keep this number handy as we move forward.

Now, why is this initial sum so important? Think of it like this: the average gives us a general idea of the numbers, but the sum gives us the total value. When we add new numbers, we’re not just changing the count; we’re changing the total value. And that change in total value directly affects the new average. So, having the original sum allows us to compare the old total with the new total and see exactly how much the 5 new numbers contributed. It’s like having a before-and-after snapshot. The 'before' is the original sum of 1950, and the 'after' will be the new sum once we add the 5 numbers. This comparison is key to solving the problem. So, always start by figuring out the initial sum when you're dealing with average problems like this. It’s the foundation for everything else.

The Impact of Adding New Numbers

Next, let’s consider the impact of adding new numbers. We’re adding 5 new numbers to our original 15, so we now have a total of 20 numbers. The average, which was 130, has increased by 60, making the new average 190. This is a significant jump, and it tells us that the 5 new numbers must have a considerable sum to pull the average up by so much. Now, to find the new total sum of these 20 numbers, we multiply the new average (190) by the new count (20). This gives us 3800. So, the sum of all 20 numbers is 3800. This is a crucial step because it shows us the total value after adding the new numbers. We’ve gone from a sum of 1950 for the original 15 numbers to a sum of 3800 for all 20 numbers. This difference is what we’re really interested in because it represents the contribution of the 5 new numbers.

But why is finding the new total sum so important? Well, think of it this way: the average is like the center of gravity for the numbers. When you add new numbers, you’re essentially shifting that center of gravity. If the new numbers are significantly higher than the old ones, the average will go up. To quantify this shift, we need to know the total weight on both sides – the original sum and the new sum. The new total sum of 3800 gives us a complete picture of the new state. It tells us the total value of all the numbers combined, which is essential for figuring out the individual contribution of the 5 new numbers. This is why we calculate the new total sum – it’s the key to unlocking the final answer. Without it, we’d be missing a crucial piece of the puzzle. So, remember, finding the new total sum is a vital step in understanding the impact of adding new numbers to an average.

Calculating the Sum of the New Numbers

Now comes the exciting part: calculating the sum of the new numbers. We know the total sum of the 20 numbers (original 15 + 5 new) is 3800, and we know the sum of the original 15 numbers is 1950. To find the sum of the 5 new numbers, we simply subtract the original sum from the new total sum. That’s 3800 minus 1950, which equals 1850. So, the sum of the 5 new numbers is 1850. This is our final answer! We’ve successfully figured out the total value of the 5 numbers that were added, which is what the problem asked us to find. It’s like we’ve solved the mystery of the missing numbers!

But let’s think about why this subtraction works. It’s all about isolating the contribution of the new numbers. We started with a total sum, added some new numbers, and got a new total sum. The difference between these two totals is exactly the sum of what we added. It’s like saying,