Calculate Differences: 3-Digit Numbers Math Problem

Hey guys! Let's dive into a fun math problem today that involves calculating differences between 3-digit numbers. We'll break it down step by step to make sure everyone understands. So, grab your pencils and let's get started!

Understanding the Problem

The problem asks us to calculate two different differences:

• Part a): Find the difference between the largest number written with three distinct digits and the smallest number written with three identical digits.
• Part b): Find the difference between the largest even number written with three distinct digits and the smallest odd number of three digits.

Before we jump into the calculations, let's make sure we understand what each part of the question means. Distinct digits mean that each digit in the number is different (e.g., 987). Identical digits mean that all the digits in the number are the same (e.g., 111). An even number is a number that can be divided by 2 without leaving a remainder, and an odd number is a number that leaves a remainder of 1 when divided by 2.

Solving Part a)

Finding the Largest Number with Three Distinct Digits

To find the largest number with three distinct digits, we need to think about which digits are the largest. The largest digit is 9, so that will be our hundreds digit. The next largest digit is 8, which will be our tens digit. And finally, the next largest digit is 7, which will be our units digit. Therefore, the largest number with three distinct digits is 987. This is a key concept in understanding number formation. We're essentially trying to maximize each place value (hundreds, tens, ones) while ensuring that no digit is repeated.

Think of it like building a tower. You want the widest base (highest hundreds digit) and then work your way down to smaller blocks (tens and units) without using the same block twice. This approach ensures you're constructing the absolutely largest possible number under the given constraints. Moreover, understanding place value is crucial in mathematics, as it allows us to effectively compare and manipulate numbers of varying sizes.

Finding the Smallest Number with Three Identical Digits

Now, let's find the smallest number with three identical digits. The smallest digit is 0, but we can't use 0 as the first digit in a 3-digit number because that would make it a 2-digit number. So, the next smallest digit is 1. Therefore, the smallest number with three identical digits is 111. This part is relatively straightforward, but it highlights the importance of considering all conditions given in a problem. We need to ensure the number is a proper 3-digit number, which rules out 0 as the leading digit.

The simplicity of this step shouldn't be overlooked. It reinforces the fundamental rules of number representation and how the position of a digit influences its value. This concept is foundational for more complex mathematical operations, such as addition, subtraction, multiplication, and division. Moreover, recognizing patterns like these can simplify problem-solving in various mathematical contexts.

Calculating the Difference

Now that we have both numbers, we can calculate the difference. The difference between 987 and 111 is:

987 - 111 = 876

So, the answer to part a) is 876.

Solving Part b)

Finding the Largest Even Number with Three Distinct Digits

For part b), we need to find the largest even number with three distinct digits. Remember, an even number ends in 0, 2, 4, 6, or 8. To make the number as large as possible, we should start with the largest digit for the hundreds place, which is 9. Then, we use the next largest digit for the tens place, which is 8. For the units place, we need an even digit. The largest even digit that hasn't been used yet is 6. Therefore, the largest even number with three distinct digits is 986.

Here, the added constraint of "even" introduces a new level of thinking. We can't simply pick the three largest digits; we need to ensure the final digit satisfies the even number condition. This problem-solving technique demonstrates how to optimize solutions while adhering to multiple criteria. It also reinforces the divisibility rule for even numbers, which is a valuable tool in number theory and arithmetic.

Finding the Smallest Odd Number of Three Digits

Next, we need to find the smallest odd number of three digits. The smallest three-digit number is 100, but that's even. The next number is 101, which is odd. So, the smallest odd number of three digits is 101.

This step requires us to consider the definition of odd numbers and the structure of three-digit numbers. It's a straightforward process of identifying the smallest number that fits both criteria. This reinforces the understanding of number sequences and the characteristics of odd and even numbers. Such concepts are fundamental building blocks in more advanced mathematical topics, such as algebra and number theory.

Calculating the Difference

Now, we calculate the difference between 986 and 101:

986 - 101 = 885

So, the answer to part b) is 885.

Final Answers

Alright guys, we've solved both parts of the problem! Here are our answers:

• a) The difference between the largest number written with three distinct digits and the smallest number written with three identical digits is 876.
• b) The difference between the largest even number written with three distinct digits and the smallest odd number of three digits is 885.

I hope this explanation was clear and helpful. Remember, the key to solving math problems is to break them down into smaller, manageable steps. Keep practicing, and you'll become a math whiz in no time! Keep up the great work, and I'll see you in the next problem-solving session! Remember practice makes perfect. Math is like a muscle, the more you exercise it, the stronger it gets. So, keep those pencils moving and those brains churning!