Solving Arithmetic Expressions A Step-by-Step Guide

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Arithmetic operations form the bedrock of mathematics, serving as essential tools for solving a myriad of problems across various disciplines. This article delves into the step-by-step solutions of two intriguing arithmetic expressions: 86,943 + 56,205 - 65,719 and 4,18,026 + 1,08,799 - 1,48,274. By meticulously dissecting each calculation, we aim to provide a comprehensive understanding of the underlying principles and techniques involved. Join us as we embark on this mathematical journey, unraveling the intricacies of these expressions and enhancing your arithmetic prowess.

a. Solving 86,943 + 56,205 - 65,719

To solve this expression, we will follow the order of operations, which dictates that addition and subtraction are performed from left to right. This ensures that we arrive at the correct answer by systematically addressing each operation.

Step 1: Addition

We begin by adding the first two numbers, 86,943 and 56,205. This involves adding the digits in each place value column, starting from the rightmost column (ones place) and moving towards the left. If the sum of the digits in a column exceeds 9, we carry over the tens digit to the next column.

  86943
+ 56205
-------
 143148

Adding the ones digits, 3 + 5 = 8. In the tens column, 4 + 0 = 4. Moving to the hundreds column, 9 + 2 = 11. We write down 1 and carry over 1 to the thousands column. In the thousands column, 6 + 6 + 1 (carry-over) = 13. We write down 3 and carry over 1 to the ten-thousands column. Finally, in the ten-thousands column, 8 + 5 + 1 (carry-over) = 14. Thus, the sum of 86,943 and 56,205 is 143,148.

Step 2: Subtraction

Now that we have the sum of the first two numbers, we subtract 65,719 from 143,148. Subtraction involves subtracting the digits in each place value column, starting from the rightmost column. If the digit in the top number is smaller than the digit in the bottom number, we borrow from the next column to the left.

 143148
- 65719
-------
  77429

Starting with the ones column, 8 - 9. Since 8 is smaller than 9, we borrow 1 from the tens column, making it 18 - 9 = 9. In the tens column, we initially had 4, but after borrowing, it becomes 3. So, we have 3 - 1 = 2. In the hundreds column, 1 - 7. Again, we need to borrow 1 from the thousands column, making it 11 - 7 = 4. In the thousands column, we had 3, but after borrowing, it becomes 2. So, we have 2 - 5. We need to borrow 1 from the ten-thousands column, making it 12 - 5 = 7. Finally, in the ten-thousands column, we initially had 14, but after borrowing, it becomes 13. So, we have 13 - 6 = 7. Therefore, the result of 143,148 - 65,719 is 77,429.

Conclusion for Part a

By performing the addition and subtraction operations step-by-step, we have successfully solved the expression 86,943 + 56,205 - 65,719, arriving at the final answer of 77,429. This meticulous approach underscores the importance of following the order of operations to ensure accuracy in arithmetic calculations. This section has meticulously demonstrated how to break down a complex arithmetic problem into manageable steps. By first adding 86,943 and 56,205 to get 143,148, and then subtracting 65,719 from the result, we arrived at the final answer of 77,429. This methodical approach ensures accuracy and clarity in problem-solving. The key to mastering such calculations lies in understanding the fundamental principles of addition and subtraction, and applying them systematically. This detailed explanation not only provides the solution but also enhances comprehension of the arithmetic process involved. Understanding these steps is crucial for tackling similar problems with confidence and precision. The step-by-step breakdown highlights the significance of each operation and its contribution to the final result. This clarity is essential for building a solid foundation in arithmetic. Practice is paramount in mastering arithmetic operations. By working through various examples, you can reinforce your understanding and improve your speed and accuracy. Remember, each problem solved is a step forward in your mathematical journey.

b. Solving 4,18,026 + 1,08,799 - 1,48,274

Similar to the previous expression, we will solve this by following the order of operations, performing addition first, followed by subtraction. This systematic approach ensures that we handle each operation correctly and arrive at the accurate result. This methodical approach is the cornerstone of arithmetic problem-solving.

Step 1: Addition

We begin by adding 4,18,026 and 1,08,799. As before, we add the digits in each place value column, starting from the ones place and moving towards the left, carrying over when necessary.

  418026
+ 108799
--------
  526825

Starting with the ones digits, 6 + 9 = 15. We write down 5 and carry over 1 to the tens column. In the tens column, 2 + 9 + 1 (carry-over) = 12. We write down 2 and carry over 1 to the hundreds column. In the hundreds column, 0 + 7 + 1 (carry-over) = 8. Moving to the thousands column, 8 + 8 = 16. We write down 6 and carry over 1 to the ten-thousands column. In the ten-thousands column, 1 + 0 + 1 (carry-over) = 2. Finally, in the lakhs column, 4 + 1 = 5. Therefore, the sum of 4,18,026 and 1,08,799 is 5,26,825.

Step 2: Subtraction

Next, we subtract 1,48,274 from 5,26,825. Again, we subtract the digits in each place value column, starting from the rightmost column, borrowing when needed.

  526825
- 148274
--------
  378551

Starting with the ones column, 5 - 4 = 1. In the tens column, 2 - 7. Since 2 is smaller than 7, we borrow 1 from the hundreds column, making it 12 - 7 = 5. In the hundreds column, we initially had 8, but after borrowing, it becomes 7. So, we have 7 - 2 = 5. In the thousands column, 6 - 8. We need to borrow 1 from the ten-thousands column, making it 16 - 8 = 8. In the ten-thousands column, we had 2, but after borrowing, it becomes 1. So, we have 1 - 4. We need to borrow 1 from the lakhs column, making it 11 - 4 = 7. Finally, in the lakhs column, we initially had 5, but after borrowing, it becomes 4. So, we have 4 - 1 = 3. Thus, the result of 5,26,825 - 1,48,274 is 3,78,551.

Conclusion for Part b

By systematically performing the addition and subtraction operations, we have successfully solved the expression 4,18,026 + 1,08,799 - 1,48,274, arriving at the final answer of 3,78,551. This step-by-step approach highlights the importance of accuracy and precision in arithmetic calculations. This detailed breakdown illustrates the process of solving a multi-step arithmetic problem. By first adding 4,18,026 and 1,08,799 to get 5,26,825, and then subtracting 1,48,274 from the result, we found the final answer to be 3,78,551. This method not only provides the answer but also reinforces the understanding of arithmetic operations. The emphasis on each step ensures that the process is clear and easy to follow. This clarity is crucial for building confidence in tackling complex calculations. The systematic approach used here can be applied to a wide range of arithmetic problems. By breaking down each problem into smaller, manageable steps, you can minimize errors and maximize accuracy. Regular practice and a thorough understanding of the fundamentals are key to excelling in arithmetic. The consistent application of these principles will enhance your problem-solving skills and mathematical proficiency.

Final Thoughts on Mastering Arithmetic Operations

In conclusion, we have meticulously solved two arithmetic expressions, 86,943 + 56,205 - 65,719 and 4,18,026 + 1,08,799 - 1,48,274, by adhering to the order of operations and performing each step with precision. The results, 77,429 and 3,78,551, respectively, demonstrate the effectiveness of a systematic approach in arithmetic calculations. Mastering arithmetic operations is not just about finding the correct answer; it's about developing a deep understanding of the underlying principles. This understanding forms the foundation for more advanced mathematical concepts and problem-solving skills. The ability to perform arithmetic operations accurately and efficiently is essential in various aspects of life, from everyday financial transactions to complex scientific calculations. The techniques and strategies discussed in this article can be applied to a wide range of arithmetic problems. By consistently practicing and applying these methods, you can enhance your mathematical skills and confidence. Remember, the key to mastering arithmetic operations is a combination of understanding the fundamentals, practicing regularly, and approaching each problem with a systematic mindset. Embrace the challenge of arithmetic, and you will unlock a world of mathematical possibilities. Arithmetic operations are the building blocks of mathematics, and mastering them is crucial for success in higher-level math courses and real-world applications. The ability to add, subtract, multiply, and divide accurately and efficiently is essential for solving a wide range of problems. Continuous practice and a solid understanding of the fundamental principles are key to achieving mastery. By breaking down complex problems into smaller, more manageable steps, you can approach them with confidence and precision. Remember, every problem solved is a step forward in your mathematical journey. Embrace the challenge and enjoy the process of learning and growing your arithmetic skills.