Which Expression Could Janice Use To Check Her Work?

To effectively check her work for the subtraction equation 3,578 - 1,542 = 2,036, Janice needs to utilize an inverse operation. The inverse operation of subtraction is addition. This principle is fundamental in mathematics and allows us to verify the accuracy of subtraction problems. In essence, if we subtract one number from another and obtain a result, adding the result back to the number we subtracted should yield the original number. This concept ensures that our calculations are correct and provides a reliable method for error checking. To illustrate, if Janice's calculation is correct, adding the difference (2,036) to the subtrahend (1,542) should result in the minuend (3,578). This method is not just a trick; it's a direct application of the relationship between addition and subtraction, rooted in the basic axioms of arithmetic. This approach helps to reinforce the understanding of number relationships and operational inverses, which are critical for more advanced mathematical concepts. By mastering this method, Janice can not only verify her answers but also deepen her understanding of how numbers interact within equations. Furthermore, this technique is broadly applicable across various mathematical contexts, making it an indispensable tool for problem-solving and accuracy assurance. For instance, in algebra, understanding inverse operations is crucial for solving equations and manipulating expressions. In more complex calculations involving multiple steps, this method can be used to check intermediate results, ensuring that errors do not propagate through the entire problem. Therefore, it is essential that Janice grasp this concept thoroughly to build a strong foundation in mathematics. Using inverse operations is not merely a way to verify answers; it is a way to ensure understanding and accuracy in mathematical thinking. This skill transcends basic arithmetic and becomes a cornerstone of mathematical proficiency.

Understanding the Inverse Relationship between Addition and Subtraction

The key to checking subtraction problems lies in understanding the inverse relationship between addition and subtraction. Addition and subtraction are operations that undo each other. This fundamental concept is crucial for verifying calculations and ensuring accuracy in mathematics. In the context of Janice’s problem, the equation 3,578 - 1,542 = 2,036 represents a subtraction operation. To check if this equation is correct, Janice can use the inverse operation, which is addition. The principle behind this method is simple yet powerful: if subtracting 1,542 from 3,578 results in 2,036, then adding 2,036 back to 1,542 should yield the original number, 3,578. This relationship forms the basis of checking subtraction through addition. By understanding this inverse relationship, Janice can confidently verify her work and identify any errors. This method also reinforces her understanding of how numbers interact within mathematical operations. It’s not just about getting the right answer; it’s about understanding why the method works, which is a crucial aspect of mathematical proficiency. Furthermore, this concept is not isolated to simple arithmetic; it extends to more advanced mathematical areas such as algebra and calculus, where inverse operations are fundamental in solving equations and simplifying expressions. Therefore, mastering the inverse relationship between addition and subtraction is a cornerstone of mathematical competence. This foundational understanding allows for more complex problem-solving and enhances the ability to approach mathematical challenges with confidence. In practical terms, this skill is invaluable for everyday calculations, financial planning, and various other real-world applications where accuracy is paramount.

Evaluating the Options

Let's analyze the given options to determine which expression Janice should use. It's important to systematically assess each option to identify the one that correctly applies the inverse operation principle. This involves understanding the structure of each expression and how it relates to the original subtraction equation. The goal is to find an expression that, when calculated, will verify whether the subtraction was performed accurately. Option A, 3,578 - 2,036 + 1,542, introduces an additional subtraction step before adding, which does not directly apply the inverse operation to check the original equation. Option B, 3,578 + 1,542, adds the minuend and the subtrahend, which doesn't utilize the result of the subtraction (2,036) to verify the original calculation. Option C, 3,578 + 1,542 + 2,036, involves adding all three numbers together, which doesn't logically check the subtraction performed. This option doesn't leverage the inverse relationship between addition and subtraction to validate the result. Option D, 1,542 + 2,036, adds the subtrahend (1,542) and the difference (2,036). This aligns perfectly with the principle of using the inverse operation to verify subtraction. If the sum of 1,542 and 2,036 equals the minuend (3,578), then the subtraction is correct. By carefully examining each option, we can see that Option D is the only one that correctly applies the concept of inverse operations to check the original subtraction equation. This methodical evaluation helps in understanding not just the answer, but also the underlying mathematical principles involved.

The Correct Expression: 1,542 + 2,036

The correct expression for Janice to check her work is D. 1,542 + 2,036. This option accurately utilizes the inverse relationship between addition and subtraction to verify the original equation. To understand why this is the correct expression, let's break it down: The original equation is 3,578 - 1,542 = 2,036. To check this, Janice needs to perform an operation that essentially reverses the subtraction. The inverse operation of subtraction is addition. Therefore, she needs to add the result of the subtraction (2,036) to the number that was subtracted (1,542). If the sum of these two numbers equals the original number (3,578), then her subtraction is correct. Mathematically, this can be expressed as 1,542 + 2,036 = 3,578. By performing this addition, Janice can directly verify her initial calculation. This method is not only effective but also reinforces the fundamental principle of inverse operations in mathematics. It’s a straightforward application of a core mathematical concept, making it an ideal way to check subtraction problems. Furthermore, this approach helps in building a deeper understanding of number relationships and operational inverses, which are essential for more advanced mathematical concepts. Therefore, by choosing option D, Janice is employing a mathematically sound strategy to confirm the accuracy of her work. This highlights the importance of understanding not just how to perform calculations, but also how to verify them using inverse operations.

Why Other Options Are Incorrect

To fully grasp why option D is the correct choice, it’s crucial to understand why the other options are incorrect. Each incorrect option presents a different mathematical operation that does not effectively verify the original subtraction equation. Option A, 3,578 - 2,036 + 1,542, involves subtracting 2,036 from 3,578 and then adding 1,542. This sequence of operations does not directly apply the inverse relationship to check the original subtraction. While it does include both subtraction and addition, it doesn't logically reverse the initial operation in a way that would confirm the result. Option B, 3,578 + 1,542, simply adds the minuend and the subtrahend. This operation doesn’t incorporate the result of the subtraction (2,036), which is necessary for verification. Adding the two original numbers doesn’t provide any information about whether the subtraction was performed correctly. Option C, 3,578 + 1,542 + 2,036, adds all three numbers together. This operation doesn’t align with the principle of inverse operations and doesn’t logically check the subtraction. Adding all the numbers will not reveal if the initial subtraction was accurate. Understanding why these options are incorrect reinforces the importance of using the correct inverse operation to verify calculations. The key takeaway is that to check a subtraction problem, you need to add the difference to the subtrahend and see if it equals the minuend. The incorrect options highlight the potential pitfalls of applying mathematical operations without a clear understanding of their purpose in the context of verification. This comprehensive analysis not only identifies the correct answer but also deepens the understanding of the mathematical principles involved.

By adding the subtrahend and the difference, Janice can effectively check her work. Therefore, the answer is D.