Finding The Range: Solving For A Between 8748 And 9948

Alright, guys, let's break down this math problem together! We're trying to figure out what number can replace 'A' in the expression 8748 < A < 9948. Basically, 'A' has to be bigger than 8748 but smaller than 9948. We need to find the biggest and smallest whole numbers that fit this rule and then find the difference between them. Think of it like finding the edges of a sweet spot! Let's dive in and make sure we nail this concept. Understanding inequalities is super important, and this is a great way to practice.

Understanding the Problem

So, first things first, let’s really understand what the problem is asking. We've got this expression: 8748 < A < 9948. This isn't an equation; it's an inequality. That little '<' symbol means 'less than'. So, we're saying that 'A' is greater than 8748 but also less than 9948.

Think of it like a number line. 8748 is on one end, and 9948 is on the other. 'A' has to fall somewhere between those two numbers, but it can't actually be either of those numbers. It's like saying you can have any piece of candy from a jar except for the first and last ones. Make sense?

The question is specifically asking for two things:

1. The largest possible number that 'A' can be.
2. The smallest possible number that 'A' can be.

Once we find those two numbers, we need to subtract the smallest from the largest to find the difference. This is a classic range-finding problem. We need to be careful with those 'less than' symbols and make sure we don't accidentally include the boundary numbers themselves.

Finding the Smallest Possible Value for A

Okay, so let's start by finding the smallest possible value for 'A'. Remember, 'A' has to be greater than 8748. That means 8748 itself doesn't work. We need to find the very next whole number. What comes right after 8748? It's 8749, right? So, 8749 is the smallest whole number that is bigger than 8748. We can check this to make sure: Is 8748 < 8749? Yes, it is!

So, the smallest possible value for A is 8749.

It might seem super obvious, but it's important to be precise, especially when dealing with inequalities. If the problem said 'A' could be greater than or equal to 8748 (written as 8748 ≤ A), then 8748 would be a possibility. But in our case, it's strictly greater than, so we have to move to the next whole number. This is a crucial detail that can change the whole answer.

Finding the Largest Possible Value for A

Alright, now let's find the largest possible value for 'A'. This time, 'A' has to be less than 9948. So, 9948 itself doesn't work. We need to find the whole number that comes right before 9948. What's one less than 9948? It's 9947, isn't it? Therefore, 9947 is the largest whole number that is smaller than 9948. Let's check it: Is 9947 < 9948? Yep, it is!

So, the largest possible value for A is 9947.

Just like before, pay close attention to the 'less than' symbol. If the problem said 'A' could be less than or equal to 9948 (written as A ≤ 9948), then 9948 would be a valid option. But since it's strictly less than, we have to take the number just before it. Spotting those little nuances in the wording is key to getting these problems right!

Calculating the Difference

Okay, we've found the smallest and largest possible values for 'A'. Now, the final step is to find the difference between them. Remember, difference means subtraction. We need to subtract the smallest value (8749) from the largest value (9947). So, the calculation is: 9947 - 8749 = ?

Let's do the subtraction:

9947

• 8749

1198

So, 9947 - 8749 = 1198.

Therefore, the difference between the largest and smallest possible values of A is 1198.

This is the answer we're looking for! Make sure to double-check your work to avoid any simple mistakes. Subtraction errors are easy to make, especially under pressure during a test. Taking a moment to review can save you from losing points!

Checking Our Answer Against the Options

Now, let's look at the answer choices given in the problem:

A) 8749 B) 9947 C) 1198 D) 1200

We calculated that the difference is 1198, which matches option C. So, option C is the correct answer. It's always a good idea to compare your answer with the provided options to make sure you haven't made a mistake along the way.

Options A and B are the smallest and largest values of A respectively, but the questions ask for the difference between these values. Option D is close to the correct answer, but it's not the correct one, probably due to a calculation error during the subtraction.

Final Answer

So, to recap, the difference between the largest and smallest possible values of A in the expression 8748 < A < 9948 is 1198. We found the smallest possible value for A to be 8749 and the largest possible value to be 9947. Subtracting the smallest from the largest gave us the difference of 1198. Make sure to check that your work is accurate and that you have answered the question that was asked. This is the last stage, so it is important to double-check it. This ensures you are not answering the wrong question or giving the incorrect numbers.

And that's how you solve it! You are all rockstars and are doing great!