Solving Math Inequality: Find The Sum Of Possible Digits
Hey guys! Let's dive into this interesting math problem together. We've got an inequality to solve, and it's all about figuring out which digits can fit into a blank space to make the statement true. It might sound a bit tricky at first, but don't worry, we'll break it down step by step. Grab your thinking caps, and let's get started!
Understanding the Problem
The problem presents us with a mathematical inequality: 66 571 + 1342 > 765 99. The goal here is to find the digit(s) that can replace the blank space (represented as an underscore) to make this inequality a true statement. Once we identify these digits, we need to calculate their sum. This involves a bit of arithmetic and logical thinking, so let's get into the nitty-gritty.
Breaking Down the Inequality
First, let's simplify the left side of the inequality by adding the numbers together:
66 571 + 1342 = 67 913
Now our inequality looks like this:
67 913 > 765 _9
We need to find the digit that, when placed in the blank space, will make the number 765_9 smaller than 67 913. This is where we start thinking about place values and how digits influence the size of a number. Remember, understanding the problem is half the solution!
The Importance of Place Value
In the number 765_9, the blank space represents the tens digit. This is a crucial point because the tens digit significantly impacts the overall value of the number. To make 765_9 smaller than 67 913, we need to carefully consider what digits can occupy that space. Let's explore the possibilities and figure out which digits fit the bill. Thinking about place value helps us narrow down the options and make smart choices.
Identifying Possible Digits
Okay, let's figure out what digits can fill in that blank. We need 765_9 to be less than 67 913. This means we need to consider the range of possible values for the tens digit.
Starting with the Basics
Let's start by thinking about the smallest possible digit: 0. If we put 0 in the blank, we get 76509. Is 76509 less than 67 913? Yes, it is! So, 0 is a valid option. Now, let's try the next digit, 1. If we put 1 in the blank, we get 76519. This is still less than 67 913. We can keep going like this, testing each digit one by one. It's a methodical way to ensure we don't miss any possibilities.
Testing Each Digit
- 0: 76509 < 67 913 (True)
- 1: 76519 < 67 913 (True)
- 2: 76529 < 67 913 (True)
- 3: 76539 < 67 913 (True)
- 4: 76549 < 67 913 (True)
- 5: 76559 < 67 913 (True)
- 6: 76569 < 67 913 (True)
So far, all the digits from 0 to 6 work. But what about 7? If we put 7 in the blank, we get 76579. Is 76579 < 67 913? No, it's not! This means that 7 and any digit larger than 7 will not work. We've found the limit! This step-by-step approach helps us confidently identify the digits that satisfy the inequality.
The Valid Digits
Based on our testing, the digits that can be written in the blank space are 0, 1, 2, 3, 4, 5, and 6. These are the heroes of our inequality! Now that we have these digits, the final step is to add them together to find the sum.
Calculating the Sum
We've identified the possible digits: 0, 1, 2, 3, 4, 5, and 6. Now, let's add them up to find the total. This is the final piece of the puzzle!
Adding the Digits Together
To find the sum, we simply add all the valid digits:
0 + 1 + 2 + 3 + 4 + 5 + 6
This might seem like a simple addition problem, but accuracy is key. A small mistake here can throw off the entire answer. So, let's take our time and add these numbers carefully.
Step-by-Step Addition
We can add these numbers in any order, but let's do it systematically:
1 + 2 = 3
3 + 3 = 6
6 + 4 = 10
10 + 5 = 15
15 + 6 = 21
So, the sum of the digits is 21. We've reached the final answer! It feels good to solve a problem from start to finish, right? This is where all our hard work pays off.
The Final Sum
The sum of the digits that can be written in the blank space to make the inequality correct is 21. That's it! We've successfully navigated this problem. Now, let's make sure we understand what this means in the context of the original question.
Conclusion
Alright, guys, we've tackled this math problem together, and we've come out on top! We started with the inequality 66 571 + 1342 > 765 _9, and we needed to find the sum of the digits that could fill the blank space. By breaking down the problem, understanding place value, and testing each digit, we found that the possible digits were 0, 1, 2, 3, 4, 5, and 6. Then, we added those digits together and discovered that their sum is 21.
Reviewing the Steps
Let's quickly recap the steps we took:
- Understood the Problem: We identified the goal of the problem and the inequality we needed to solve.
- Simplified the Inequality: We added 66 571 and 1342 to get 67 913.
- Identified Possible Digits: We tested digits 0 through 9 to see which ones would make the inequality true.
- Calculated the Sum: We added the valid digits together.
By following these steps, we were able to systematically solve the problem and arrive at the correct answer. This kind of step-by-step approach is super helpful for tackling all sorts of math challenges.
The Final Answer
So, the final answer is 21. If you encountered this problem on a test or assignment, you'd confidently select the option that corresponds to 21. Great job, everyone! We've conquered this math mountain together!
Why This Matters
Understanding how to solve inequalities and work with place value isn't just about getting the right answer on a test. These skills are fundamental to many areas of math and science. They help us make comparisons, understand relationships between numbers, and solve complex problems. The ability to break down a problem into smaller, manageable steps is a valuable life skill, too. These problem-solving skills are super handy in everyday life.
Keep Practicing!
If you enjoyed this problem, keep practicing! The more you work with inequalities and mathematical expressions, the more comfortable and confident you'll become. Try finding similar problems online or in textbooks, and challenge yourself to solve them. Remember, every problem you solve is a step forward in your mathematical journey.
So, keep up the great work, guys! You've got this! And remember, math can be fun when we approach it with curiosity and a willingness to learn. Until next time, happy problem-solving!