Dividing Mixed Numbers A Step-by-Step Guide For -12 1/8 ÷ 1 1/2

July 13, 2025 by Scholario Team 64 views

$Dividing Mixed Numbers: A Comprehensive Guide to Solving $-12 \frac{1}{8} \div 1 \frac{1}{2}$$

Mathematics often presents us with intriguing challenges, and dividing mixed numbers is a fundamental concept that requires a thorough understanding. In this comprehensive guide, we will delve deep into the process of dividing mixed numbers, using the example $-12 \frac{1}{8} \div 1 \frac{1}{2}$ as our focal point. We will break down each step, ensuring clarity and fostering a strong grasp of the underlying principles. Mastering this skill is crucial for various mathematical applications, from everyday problem-solving to advanced algebraic manipulations.

Understanding Mixed Numbers

Before we embark on the division process, it is essential to understand what mixed numbers are and how they function. A mixed number is a combination of a whole number and a proper fraction. In our example, $-12 \frac{1}{8}$ and $1 \frac{1}{2}$ are mixed numbers. The whole number part represents the number of complete units, while the fractional part represents a portion of a unit. For instance, $-12 \frac{1}{8}$ signifies negative twelve whole units plus one-eighth of another unit. The mixed number $1 \frac{1}{2}$ represents one whole unit and one-half of another unit. Recognizing the composition of mixed numbers is the first step towards performing operations like division effectively.

Converting Mixed Numbers to Improper Fractions

The cornerstone of dividing mixed numbers lies in converting them into improper fractions. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This conversion is crucial because it allows us to apply the rules of fraction division seamlessly. To convert a mixed number to an improper fraction, we follow a simple procedure:

Multiply the whole number by the denominator of the fractional part.
Add the numerator of the fractional part to the result.
Place the sum over the original denominator.

Let's apply this to our example. For $-12 \frac{1}{8}$ , we multiply -12 by 8, which gives us -96. Then, we add the numerator 1, resulting in -95. Placing this over the original denominator 8, we get the improper fraction $-\frac{97}{8}$ . Similarly, for $1 \frac{1}{2}$ , we multiply 1 by 2, which gives us 2. Adding the numerator 1 results in 3. Placing this over the denominator 2, we get the improper fraction $\frac{3}{2}$ . It's important to handle the negative sign correctly when dealing with negative mixed numbers; it remains with the improper fraction.

The Division Process: Dividing Fractions

Once we have converted the mixed numbers into improper fractions, we can proceed with the division. Dividing fractions involves a simple yet crucial step: multiplying by the reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$ . To divide fractions, we invert the second fraction (the divisor) and then multiply the two fractions.

Applying the Reciprocal

In our example, we have $-\frac{97}{8} \div \frac{3}{2}$ . To divide these fractions, we need to find the reciprocal of $\frac{3}{2}$ , which is $\frac{2}{3}$ . Now, we rewrite the division problem as a multiplication problem: $-\frac{97}{8} \times \frac{2}{3}$ . This transformation is the key to dividing fractions efficiently. By changing division to multiplication using the reciprocal, we simplify the process and make it easier to arrive at the correct answer. Understanding this step is fundamental to mastering fraction division.

Multiplying Fractions

After applying the reciprocal, we are left with a multiplication problem involving fractions. To multiply fractions, we multiply the numerators together and the denominators together. In our case, we have $-\frac{97}{8} \times \frac{2}{3}$ . Multiplying the numerators, we get $-97 \times 2 = -194$ . Multiplying the denominators, we get $8 \times 3 = 24$ . Therefore, the result of the multiplication is $-\frac{194}{24}$ . This fraction represents the quotient of our original division problem, but it is not yet in its simplest form. Simplification is the next crucial step in the process.

Simplifying Improper Fractions

Now that we have the improper fraction $-\frac{194}{24}$ , we need to simplify it. Simplifying a fraction means reducing it to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by the GCD. In some cases, it may also involve converting the improper fraction back into a mixed number for easier interpretation. Simplification ensures that our answer is presented in the most concise and understandable form.

Finding the Greatest Common Divisor (GCD)

To simplify $-\frac{194}{24}$ , we need to find the greatest common divisor (GCD) of 194 and 24. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. One way to find the GCD is by listing the factors of each number and identifying the largest factor they have in common. Alternatively, we can use the Euclidean algorithm, which is a more systematic approach for finding the GCD of two numbers. For 194 and 24, the GCD is 2. This means that 2 is the largest number that divides both 194 and 24 evenly. Identifying the GCD is a critical step in simplifying fractions effectively.

Dividing by the GCD

Having found the GCD, we now divide both the numerator and the denominator of $-\frac{194}{24}$ by 2. Dividing 194 by 2 gives us 97, and dividing 24 by 2 gives us 12. Thus, our simplified fraction is $-\frac{97}{12}$ . This fraction is in its simplest form because 97 and 12 have no common factors other than 1. However, it is still an improper fraction, meaning the numerator is larger than the denominator. For many applications, it is preferable to convert an improper fraction into a mixed number for easier understanding and interpretation.

Converting Back to a Mixed Number

Finally, we convert the simplified improper fraction $-\frac{97}{12}$ back into a mixed number. To do this, we divide the numerator (97) by the denominator (12). 97 divided by 12 is 8 with a remainder of 1. The quotient, 8, becomes the whole number part of the mixed number. The remainder, 1, becomes the numerator of the fractional part, and the denominator remains 12. Therefore, $-\frac{97}{12}$ is equal to $-8 \frac{1}{12}$ . This final conversion provides us with a clear and intuitive representation of the result of our division problem.

Conclusion

In conclusion, dividing mixed numbers involves a series of steps: converting mixed numbers to improper fractions, multiplying by the reciprocal, simplifying the resulting fraction, and, if necessary, converting back to a mixed number. By following these steps carefully and systematically, we can confidently tackle division problems involving mixed numbers. Our example, $-12 \frac{1}{8} \div 1 \frac{1}{2}$ , illustrates the process, culminating in the answer $-8 \frac{1}{12}$ . This method is not only applicable to this specific problem but also forms a solid foundation for more complex mathematical operations involving fractions and mixed numbers. Mastering these skills opens doors to a deeper understanding of mathematics and its applications in various fields.