Math Problem: Finding 'a' And Calculating The Result

Let's break down this math problem step by step. Guys, it looks like we're dealing with inequalities and finding numbers within a certain range. Then, we need to do some addition and subtraction. So, let's put on our math hats and get started!

Understanding the Inequality: 2578 < a < 3756

The problem states that 'a' is greater than 2578 and less than 3756. This means 'a' can be any number between these two values, but it cannot be 2578 or 3756 itself. We need to find the largest and smallest whole numbers that 'a' can be.

Finding the Smallest Possible Value of 'a'

The smallest whole number greater than 2578 is 2579. So, the smallest possible value for 'a' is 2579. Think of it like climbing stairs; if you're above the 2578th stair, the next whole stair you're on is the 2579th.

Finding the Largest Possible Value of 'a'

The largest whole number less than 3756 is 3755. Therefore, the largest possible value for 'a' is 3755. Similarly, if you're below the 3756th stair, the last whole stair you were on is the 3755th.

Calculating the Sum of the Largest and Smallest Values of 'a'

Now that we know the smallest and largest possible values of 'a', we need to add them together:

2579 + 3755 = 6334

So, the sum of the largest and smallest possible values of 'a' is 6334.

Identifying the Smallest 4-Digit Number with Distinct Digits

We need to find the smallest 4-digit number where all the digits are different. Let's think about this. The smallest 4-digit number is 1000, but it has repeating digits (three 0s). To make it the smallest with distinct digits, we want to keep the thousands digit as small as possible. So, let's start with 1.

The next digit (hundreds) should also be as small as possible, but it has to be different from 1. So, we'll use 0.

For the tens digit, we want the smallest digit that's different from 1 and 0, which is 2.

Finally, for the units digit, we want the smallest digit that's different from 1, 0, and 2, which is 3.

Therefore, the smallest 4-digit number with distinct digits is 1023.

Performing the Subtraction

The problem asks us to subtract the smallest 4-digit number with distinct digits (1023) from the sum of the largest and smallest values of 'a' (6334).

6334 - 1023 = 5311

So, the final result is 5311.

Final Answer

Therefore, the answer to the question is 5311. We found the smallest and largest possible values for 'a' based on the given inequality, calculated their sum, identified the smallest 4-digit number with distinct digits, and then performed the subtraction. Nice job, everyone!

Why This Problem Matters

You might be wondering, "Why do I need to know this?" Well, these types of problems help you develop important mathematical and logical thinking skills. Here's a breakdown:

• Understanding Inequalities: Inequalities are used everywhere in math and science. Learning to interpret and work with them is crucial. Think about setting limits on a variable in a computer program, or understanding the range of acceptable values in an experiment.

• Logical Reasoning: Finding the smallest 4-digit number with distinct digits requires logical thinking and a systematic approach. You have to consider the constraints (4 digits, all different) and work your way through the possibilities.

• Problem-Solving: Breaking down a complex problem into smaller, manageable steps is a valuable skill in any field. This problem forces you to do just that – understand the inequality, find the values, identify the number, and then perform the calculation.

• Attention to Detail: Math problems often hinge on small details. Missing the "distinct digits" requirement would lead to the wrong answer. This emphasizes the importance of reading carefully and paying attention to the specifics.

Real-World Applications

While this specific problem might not come up in your daily life, the skills it develops are highly applicable:

• Computer Programming: Setting variable limits, validating data, and writing algorithms all rely on understanding inequalities and logical reasoning.

• Finance: Calculating interest rates, analyzing investment risks, and budgeting all involve working with numbers and constraints.

• Science and Engineering: Understanding experimental ranges, analyzing data, and designing systems require mathematical and logical thinking.

• Everyday Life: Even simple tasks like planning a trip (considering distance and time) or managing your finances (budgeting and saving) involve problem-solving skills similar to those used in this problem.

So, while this problem might seem abstract, it's actually building a foundation for many real-world applications. Keep practicing, and you'll see how these skills come in handy!

Practice Problems

Want to test your skills further? Try these practice problems:

1. Find the largest and smallest integer values of 'b' such that 1250 < b < 2375. Then, subtract the smallest 3-digit number with distinct digits from their sum.

2. What is the result when the sum of the largest and smallest possible values of 'x' (where 5678 < x < 6890) is subtracted from the largest 4-digit number with distinct digits?

3. 'c' is a number between 3456 and 4567. Find the difference between the largest and smallest possible values of 'c', and then add the result to the smallest 4-digit number with repeating digits.

By working through these problems, you can solidify your understanding of inequalities, logical reasoning, and problem-solving techniques. Good luck, and have fun! Remember, the key is to break down each problem into smaller steps and think systematically. You got this! And if you are still having a hard time understanding it, seek the help of your teacher.