Solving Math Expressions A Step-by-Step Guide

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In mathematics, translating word problems into mathematical expressions is a fundamental skill. This article will guide you through converting verbal phrases into mathematical equations and then solving them. We'll cover examples involving division and negative numbers, offering step-by-step solutions and explanations to enhance your understanding.

1. The Opposite of 210 over 70

To solve mathematical expressions, we need to break down the phrase "the opposite of 210 over 70" step by step. First, let's address "210 over 70." In mathematical terms, "over" often indicates division. Thus, "210 over 70" can be written as the fraction 210/70 or the division problem 210 ÷ 70. The phrase "the opposite of" implies that we need to find the negative of the result of this division. This is a crucial concept in understanding number lines and the relationship between positive and negative numbers.

Now, let's perform the division. 210 divided by 70 equals 3. So, 210/70 = 3. However, the original phrase asks for the opposite of this result. The opposite of a number is its negative counterpart. For instance, the opposite of 5 is -5, and the opposite of -3 is 3. Therefore, the opposite of 3 is -3. Hence, the mathematical expression for "the opposite of 210 over 70" is -(210/70), and the solution is -3. Understanding the concept of opposites is essential, especially when dealing with mathematical concepts such as absolute values and inequalities. Remember, the opposite of a positive number is negative, and the opposite of a negative number is positive. In this case, we began with a division that resulted in a positive number (3), and by taking the opposite, we arrived at a negative number (-3). This simple example illustrates the importance of careful interpretation and step-by-step problem-solving in mathematics.

  • Mathematical Expression: -(210/70)
  • Solution: -3

2. Negative 4,200 Divided by 300

When confronted with "negative 4,200 divided by 300," the initial step involves understanding that a negative number is being divided by a positive number. In mathematical terms, this can be written as -4200 ÷ 300 or -4200/300. The presence of a negative sign is critical, as it will significantly impact the final result. Division involving negative numbers follows specific rules: when a negative number is divided by a positive number, the quotient is negative. Conversely, when a negative number is divided by a negative number, the quotient is positive. These rules are fundamental in arithmetic and algebra.

Next, let’s perform the division. 4,200 divided by 300 is 14. However, since we are dividing a negative number by a positive number, the result will be negative. Thus, -4200 ÷ 300 = -14. To ensure accuracy, it can be helpful to check the answer by multiplying the quotient (-14) by the divisor (300). The result should be the original dividend (-4200). This verification step is a useful practice in mathematics to minimize errors and build confidence in the solution. Furthermore, understanding division with negative numbers is essential for various mathematical applications, including financial calculations, temperature measurements, and more complex algebraic equations. This example illustrates the importance of paying close attention to signs and adhering to the rules of arithmetic when solving mathematical problems.

  • Mathematical Expression: -4200 ÷ 300
  • Solution: -14

3. Negative 50 Divided by Positive 10

To tackle "negative 50 divided by positive 10," we recognize that this is another instance of dividing a negative number by a positive number. This can be mathematically represented as -50 ÷ 10 or -50/10. As with the previous example, the negative sign plays a crucial role in determining the sign of the final answer. Remember, in division, when a negative number is divided by a positive number, the result is always negative. This is a fundamental rule in arithmetic and is vital for solving problems accurately.

Now, let’s perform the division. 50 divided by 10 is 5. However, because we are dividing a negative number (-50) by a positive number (10), the quotient will be negative. Therefore, -50 ÷ 10 = -5. Just as before, it’s a good practice to check the solution by multiplying the quotient (-5) by the divisor (10). The result should equal the dividend (-50). This step ensures the correctness of the calculation and helps reinforce the understanding of the division process. This type of problem is a building block for more advanced mathematical concepts, including algebraic expressions and equations. Understanding the rules of division involving negative numbers is crucial for success in these areas.

  • Mathematical Expression: -50 ÷ 10
  • Solution: -5

4. 54 Divided by 27

For the phrase "54 divided by 27," we are dealing with the division of two positive numbers. This is a straightforward division problem, and the mathematical expression can be written as 54 ÷ 27 or 54/27. Since both numbers are positive, we can expect the result to be positive as well. This situation is less complex than the previous examples involving negative numbers, but it still provides an opportunity to reinforce the basic principles of division.

Now, let’s carry out the division. 54 divided by 27 is 2. Therefore, 54 ÷ 27 = 2. In this case, the result is a whole number, which simplifies the solution. To verify the answer, we can multiply the quotient (2) by the divisor (27). The product should be equal to the dividend (54). This verification step is a useful habit to develop in mathematics, as it helps to confirm the accuracy of the solution and build confidence in problem-solving skills. This example, though simple, underscores the foundational nature of division in mathematics and its applicability in numerous real-world scenarios.

  • Mathematical Expression: 54 ÷ 27
  • Solution: 2

Conclusion

Translating words into mathematical expressions is a crucial skill in mathematics. By carefully dissecting the phrases and applying the correct operations, we can solve a variety of problems. These examples demonstrate how to handle division, negative numbers, and the concept of opposites. Remember to always double-check your work and practice regularly to improve your mathematical proficiency.