Multiplying Fractions How To Simplify -1/6 * 5/7
In the realm of mathematics, mastering the art of multiplying fractions is a fundamental skill that unlocks a world of possibilities. Whether you're a student grappling with homework or an adult navigating everyday calculations, understanding how to multiply fractions and simplify the result is essential. In this comprehensive guide, we will delve into the process of multiplying the fractions -1/6 and 5/7, providing a clear, step-by-step explanation that will empower you to confidently tackle similar problems. Our primary focus will be on ensuring that the final answer is presented in its simplest form, a crucial aspect of mathematical precision and elegance. This detailed exploration will not only equip you with the procedural knowledge but also enhance your understanding of the underlying mathematical principles, enabling you to apply these skills in various contexts. By the end of this guide, you will be proficient in multiplying fractions, simplifying results, and appreciating the beauty of mathematical simplicity.
Before we dive into the specific problem of multiplying -1/6 and 5/7, let's establish a solid foundation by revisiting the fundamental principles of fraction multiplication. At its core, multiplying fractions involves a straightforward process: multiplying the numerators (the top numbers) together and then multiplying the denominators (the bottom numbers) together. This method applies universally, regardless of whether the fractions are positive, negative, or a combination thereof. Understanding this basic principle is the cornerstone of success in fraction multiplication. However, the journey doesn't end with simply multiplying the numerators and denominators. The next crucial step is simplification. Simplification is the process of reducing a fraction to its lowest terms, where the numerator and denominator have no common factors other than 1. This ensures that the answer is presented in its most concise and easily interpretable form. In this guide, we will emphasize the importance of both multiplication and simplification, providing you with a complete understanding of the process. We will also address the nuances of handling negative fractions, ensuring that you are well-equipped to tackle any fraction multiplication problem that comes your way. So, let's embark on this mathematical journey, armed with a clear understanding of the basic principles and a commitment to mastering the art of fraction multiplication.
Now, let's apply these principles to the specific problem at hand: multiplying the fractions -1/6 and 5/7. This step-by-step solution will not only provide the answer but also illustrate the process in a clear and understandable manner. The first step, as we've established, is to multiply the numerators. In this case, the numerators are -1 and 5. Multiplying these together, we get -1 * 5 = -5. This result becomes the numerator of our new fraction. Next, we move on to the denominators. The denominators in our problem are 6 and 7. Multiplying these together, we get 6 * 7 = 42. This result becomes the denominator of our new fraction. So, after multiplying the numerators and denominators, we arrive at the fraction -5/42. But our journey doesn't end here. The final crucial step is to determine whether this fraction can be simplified further. To do this, we need to identify the greatest common factor (GCF) of the numerator and denominator. If the GCF is 1, then the fraction is already in its simplest form. If the GCF is greater than 1, we need to divide both the numerator and denominator by the GCF to simplify the fraction. In the next section, we will delve into the process of simplification and determine whether -5/42 can be reduced to its lowest terms. This step-by-step approach ensures that we not only arrive at the correct answer but also understand the reasoning behind each step, fostering a deeper understanding of fraction multiplication.
Step 1: Multiply the Numerators
The initial step in multiplying fractions, as we've previously discussed, involves focusing on the numerators, which are the numbers positioned at the top of each fraction. In our specific problem, we are tasked with multiplying -1/6 and 5/7. Identifying the numerators in these fractions, we have -1 from the first fraction and 5 from the second fraction. The operation we need to perform is the multiplication of these two numbers: -1 multiplied by 5. This is a straightforward multiplication problem involving a negative number and a positive number. When multiplying a negative number by a positive number, the result is always a negative number. In this instance, -1 multiplied by 5 equals -5. This result, -5, now becomes the numerator of our product fraction. It's crucial to pay close attention to the signs of the numbers being multiplied, as this directly impacts the sign of the final result. A simple rule to remember is that multiplying numbers with different signs (one positive and one negative) yields a negative result, while multiplying numbers with the same sign (both positive or both negative) yields a positive result. With the numerator of our product fraction successfully determined, we can now proceed to the next step: multiplying the denominators. This methodical approach ensures that we handle each part of the fraction multiplication process with precision and accuracy.
Step 2: Multiply the Denominators
Having successfully multiplied the numerators, we now turn our attention to the denominators, which are the numbers located at the bottom of each fraction. In the problem we're addressing, multiplying -1/6 and 5/7, the denominators are 6 and 7. Our task in this step is to multiply these two numbers together. This is a fundamental multiplication operation: 6 multiplied by 7. The result of this multiplication is 42. This value, 42, becomes the denominator of our product fraction. With both the numerator and the denominator now calculated, we have successfully performed the initial multiplication of the two fractions. At this stage, we have the fraction -5/42. However, our work is not yet complete. A crucial aspect of working with fractions is ensuring that the final answer is presented in its simplest form. This means reducing the fraction to its lowest terms, where the numerator and denominator have no common factors other than 1. In the next step, we will delve into the process of simplifying fractions and determine whether -5/42 can be further reduced. This step is essential for presenting a mathematically sound and elegant answer. So, let's proceed to the simplification process, armed with our understanding of the basic principles of fraction reduction.
Step 3: Simplify the Result
After multiplying the numerators and denominators, we arrived at the fraction -5/42. Now, the crucial step is to simplify this fraction to its lowest terms. Simplification involves identifying the greatest common factor (GCF) of the numerator and the denominator and then dividing both by this GCF. The GCF is the largest number that divides both the numerator and denominator without leaving a remainder. In our case, the numerator is -5 and the denominator is 42. To find the GCF, we can list the factors of each number. The factors of 5 are 1 and 5. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. Comparing the factors of 5 and 42, we can see that the only common factor is 1. This means that the greatest common factor (GCF) of 5 and 42 is 1. When the GCF is 1, it indicates that the fraction is already in its simplest form. There are no other common factors that can be divided out to further reduce the fraction. Therefore, the fraction -5/42 is indeed in its simplest form. This concludes our simplification process. We have successfully determined that the fraction cannot be reduced further, and we can confidently present -5/42 as our final answer. Understanding the concept of GCF and the process of simplification is paramount in working with fractions, ensuring that the answers are presented in their most concise and mathematically accurate form. In this instance, our diligent effort to simplify has confirmed that -5/42 is the simplest representation of the product of -1/6 and 5/7.
Having meticulously walked through the process of multiplying the fractions -1/6 and 5/7, we have arrived at the final answer in its simplest form. Our journey began with understanding the fundamental principles of fraction multiplication, where we learned that multiplying fractions involves multiplying the numerators together and the denominators together. We then applied this principle to our specific problem, multiplying -1 by 5 to get -5 as the numerator and 6 by 7 to get 42 as the denominator, resulting in the fraction -5/42. However, we didn't stop there. Recognizing the importance of presenting answers in their simplest form, we embarked on the crucial step of simplification. We delved into the concept of the greatest common factor (GCF) and determined that the GCF of 5 and 42 is 1. This revelation confirmed that the fraction -5/42 is already in its simplest form, as there are no common factors other than 1 that can be divided out. Therefore, with confidence and mathematical precision, we can declare that the final answer to the multiplication of -1/6 and 5/7 is -5/42. This result represents the culmination of our step-by-step approach, highlighting the significance of both multiplication and simplification in the realm of fractions. By understanding and applying these principles, you are well-equipped to tackle a wide range of fraction-related problems.
In conclusion, the process of multiplying fractions, as demonstrated through the example of -1/6 * 5/7, involves a systematic approach that encompasses both multiplication and simplification. The journey begins with a firm grasp of the fundamental principles of fraction multiplication: multiplying the numerators together and the denominators together. This initial step yields a fraction that represents the product of the two original fractions. However, the process doesn't end there. The crucial step of simplification ensures that the final answer is presented in its most concise and mathematically elegant form. Simplification involves identifying the greatest common factor (GCF) of the numerator and the denominator and then dividing both by the GCF. If the GCF is 1, as in our example with -5/42, the fraction is already in its simplest form. Through this detailed exploration, we have not only solved the specific problem but also reinforced the importance of understanding the underlying mathematical principles. The ability to multiply and simplify fractions is a fundamental skill that transcends the classroom, finding applications in everyday life and various professional fields. By mastering this skill, you empower yourself to confidently navigate a wide range of mathematical challenges. So, embrace the principles we've discussed, practice diligently, and revel in the satisfaction of arriving at accurate and simplified solutions.
