Solve The Math Problem: -(32/25) ÷ (-18/15).
Introduction
In this article, we will delve into the process of solving the mathematical expression -(32/25) ÷ (-18/15). This problem involves the division of two negative fractions, and understanding the steps to solve it is crucial for mastering basic arithmetic operations. We will break down the problem into manageable steps, explaining each operation in detail to ensure clarity. Whether you are a student learning about fractions or someone looking to refresh your math skills, this guide will provide a comprehensive understanding of how to solve this type of problem efficiently and accurately.
Understanding the Basics of Fraction Division
Before we dive into the specifics of this problem, let’s review the fundamental principles of dividing fractions. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of 2/3 is 3/2. This concept is essential for simplifying division problems involving fractions. Furthermore, it’s important to remember the rules for multiplying and dividing signed numbers. A negative number divided by a negative number results in a positive number, which is a key aspect of solving our problem.
Step-by-Step Solution
Now, let’s tackle the problem -(32/25) ÷ (-18/15) step by step. First, we need to understand that dividing by a fraction is the same as multiplying by its reciprocal. So, we will rewrite the division as a multiplication problem. The reciprocal of -18/15 is -15/18. Thus, our problem becomes -(32/25) × (-15/18). Next, we multiply the numerators together and the denominators together. This gives us (32 × 15) / (25 × 18). Before performing the multiplication, we can simplify the fractions by finding common factors. 32 and 18 share a common factor of 2, and 15 and 25 share a common factor of 5. Simplifying these factors makes the calculation easier. After simplification, we multiply the remaining factors to get our final result. This methodical approach ensures accuracy and clarity in the solution process.
Step 1: Rewrite the Division as Multiplication
The initial step in solving the expression -(32/25) ÷ (-18/15) involves transforming the division operation into multiplication. This is a fundamental principle in fraction arithmetic. To divide by a fraction, we multiply by its reciprocal. The reciprocal of a fraction is obtained by inverting the numerator and the denominator. In this case, we need to find the reciprocal of -18/15. By swapping the numerator and the denominator, we get -15/18. This transformation is crucial because it simplifies the problem and makes it easier to solve. The original expression can now be rewritten as a multiplication problem, which is -(32/25) × (-15/18). This step is essential for setting up the problem for further simplification and calculation.
Understanding Reciprocals
To fully grasp this step, it’s important to understand the concept of reciprocals. A reciprocal, also known as the multiplicative inverse, is a number which, when multiplied by the original number, equals 1. For a fraction a/b, the reciprocal is b/a. When we divide by a fraction, we are essentially asking how many times that fraction fits into the dividend. Multiplying by the reciprocal gives us a more straightforward way to answer this question. In our problem, the reciprocal of -18/15 is -15/18. This means that dividing by -18/15 is the same as multiplying by -15/18. This principle is not just a rule to memorize; it is a fundamental concept in arithmetic that simplifies many types of problems.
The Significance of Changing Division to Multiplication
The transformation from division to multiplication is significant because multiplication is often easier to handle, especially when dealing with fractions. Multiplication allows us to simply multiply the numerators together and the denominators together. This is a more direct operation compared to division, which requires an additional step of finding the reciprocal. By changing the division to multiplication, we reduce the complexity of the problem and make it more manageable. This step also aligns with the order of operations, which dictates that multiplication and division should be performed from left to right. Therefore, converting the division to multiplication is not just a convenience; it is a necessary step for solving the problem correctly and efficiently. By understanding the underlying principles and the significance of this transformation, we can approach similar problems with confidence and accuracy.
Step 2: Multiply the Fractions
Having rewritten the division as multiplication, the next step is to multiply the fractions: -(32/25) × (-15/18). To multiply fractions, we multiply the numerators together and the denominators together. This means we will multiply 32 by 15 and 25 by 18. The expression now becomes (32 × 15) / (25 × 18). Before we perform the actual multiplication, it’s often beneficial to look for opportunities to simplify the fractions. Simplifying before multiplying can make the calculations easier and reduce the risk of errors. This step is crucial for arriving at the correct answer in the most efficient way. By breaking down the multiplication process, we can handle the problem systematically and accurately.
Numerator and Denominator Multiplication
The process of multiplying fractions involves combining the numerators and the denominators separately. The product of the numerators becomes the new numerator, and the product of the denominators becomes the new denominator. In our case, we have 32 multiplied by 15 as the new numerator and 25 multiplied by 18 as the new denominator. This straightforward approach makes fraction multiplication relatively simple. However, the resulting numbers can sometimes be quite large, which is why simplification before multiplication is highly recommended. By understanding this basic principle of fraction multiplication, we can confidently tackle more complex problems involving fractions.
The Importance of Simplifying Before Multiplying
Simplifying fractions before multiplying is a strategic step that can significantly reduce the complexity of the calculations. Large numbers can be cumbersome to work with, and simplifying helps to avoid potential errors. Simplification involves finding common factors between the numerators and the denominators and dividing both by these factors. In our problem, we can see common factors between 32 and 18, and between 15 and 25. By identifying and canceling out these common factors, we reduce the size of the numbers we need to multiply, making the process much more manageable. This practice not only saves time but also increases the accuracy of the final result. Therefore, simplifying before multiplying is a best practice in fraction arithmetic.
Step 3: Simplify the Fractions
Before we proceed with multiplying the numerators and denominators in the expression (32 × 15) / (25 × 18), it's highly advantageous to simplify the fractions. Simplifying fractions involves identifying common factors between the numerators and denominators and reducing them to their lowest terms. This process makes the subsequent multiplication much easier and less prone to errors. In this specific case, we can identify common factors between 32 and 18, as well as between 15 and 25. Simplifying these fractions before multiplying will significantly streamline the calculation process and ensure accuracy in our final result. By taking the time to simplify, we make the problem more manageable and reduce the risk of making mistakes with larger numbers. This step is a critical component of efficient fraction manipulation.
Identifying Common Factors
To effectively simplify fractions, it's essential to identify the common factors between the numerators and denominators. A common factor is a number that divides evenly into both the numerator and the denominator. In our expression (32 × 15) / (25 × 18), we can observe that 32 and 18 share a common factor of 2. Similarly, 15 and 25 share a common factor of 5. Recognizing these common factors is the first step towards simplification. Once we identify these factors, we can divide both the numerator and the denominator by the common factor, reducing the fraction to its simplest form. This step is crucial for making the multiplication process more straightforward and less cumbersome.
The Process of Simplification
The process of simplification involves dividing both the numerator and the denominator by their common factors. For instance, 32 and 18 have a common factor of 2. Dividing 32 by 2 gives us 16, and dividing 18 by 2 gives us 9. So, 32/18 simplifies to 16/9. Similarly, 15 and 25 have a common factor of 5. Dividing 15 by 5 gives us 3, and dividing 25 by 5 gives us 5. Thus, 15/25 simplifies to 3/5. By performing these simplifications, our expression now looks like (16 × 3) / (5 × 9). Notice how the numbers have become smaller and easier to work with. This simplification step is not just about reducing the numbers; it's about making the overall calculation more manageable and less prone to errors. By simplifying fractions before multiplying, we can ensure a more efficient and accurate solution.
Step 4: Perform the Final Calculation
After simplifying the fractions, we arrive at the simplified expression: (16 × 3) / (5 × 9). Now, we can proceed with the final calculation by multiplying the remaining numerators and denominators. Multiplying 16 by 3 gives us 48, and multiplying 5 by 9 gives us 45. So, our fraction becomes 48/45. However, we're not quite done yet. It's essential to check if the resulting fraction can be further simplified. In this case, 48 and 45 share a common factor of 3. Dividing both 48 and 45 by 3 will give us the simplest form of the fraction. This final simplification ensures that we have the most accurate and reduced answer. By performing this step-by-step calculation, we can confidently arrive at the solution.
Multiplying the Simplified Numerators and Denominators
The core of this step involves performing the multiplication of the simplified numerators and denominators. With the expression (16 × 3) / (5 × 9), we multiply 16 by 3 to get 48, and 5 by 9 to get 45. This results in the fraction 48/45. This step is a direct application of the multiplication rule for fractions, where the product of the numerators becomes the new numerator, and the product of the denominators becomes the new denominator. While this multiplication is straightforward, the resulting fraction may not be in its simplest form, which is why the next part of this step is crucial. By accurately multiplying the simplified components, we set the stage for the final reduction and ensure the correctness of our solution.
Final Simplification for the Result
After obtaining the fraction 48/45, the final step is to ensure that it is in its simplest form. This involves identifying any common factors between the numerator and the denominator and dividing both by these factors. In this case, 48 and 45 share a common factor of 3. To simplify, we divide both 48 and 45 by 3. Dividing 48 by 3 gives us 16, and dividing 45 by 3 gives us 15. Thus, the simplified fraction is 16/15. This final simplification is essential because it presents the answer in its most reduced and easily understandable form. A fraction is considered fully simplified when the numerator and denominator have no common factors other than 1. By completing this final simplification, we ensure the accuracy and clarity of our answer.
Final Answer
After following all the steps outlined above, we have successfully solved the problem -(32/25) ÷ (-18/15). We began by rewriting the division as multiplication, then multiplied the fractions, simplified where possible, and performed the final calculations. The simplified fraction we arrived at is 16/15. It's also important to note that since we were dividing a negative number by a negative number, the result is positive. Therefore, the final answer is 16/15. This result represents the solution to the original mathematical expression and showcases the effectiveness of breaking down a problem into manageable steps.
Verification of the Solution
To ensure the accuracy of our solution, it's a good practice to verify the result. One way to do this is to reverse the steps and check if we arrive back at the original expression. We found that -(32/25) ÷ (-18/15) = 16/15. To verify, we can multiply 16/15 by -18/15 (the original divisor) and see if we get -32/25 (the original dividend). This check involves performing the multiplication and simplification steps in reverse. If the verification yields the original dividend, it confirms the correctness of our solution. This process of verification adds confidence to our answer and reinforces the understanding of the mathematical operations involved.
Implications of a Positive Result
The final answer of 16/15 is a positive fraction, which is a significant aspect of the solution. This positivity arises from the fact that we were dividing a negative number by another negative number. In mathematics, a negative number divided by a negative number always results in a positive number. This principle is fundamental in understanding the behavior of signed numbers in arithmetic operations. The positive result of 16/15 is a direct consequence of this rule. It's essential to keep track of the signs when performing calculations to ensure the final answer has the correct sign. The positive result not only validates the numerical value but also confirms the adherence to mathematical rules regarding signed numbers. By understanding these implications, we gain a deeper insight into the problem and its solution.