Problem 72 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$-\frac{5}{16} \div \frac{25}{8}$$

Step-by-Step Solution

Answer

The simplified result of the division operation \(-\frac{5}{16} \div \frac{25}{8}\) is \(-\frac{1}{10}\)

Step 1: Identify the problem

To solve the problem, you need to perform the division operation between two fractions, \(-\frac{5}{16}\) and \(\frac{25}{8}\)

Step 2: Compute the reciprocal of the divisor

The reciprocal of a fraction is obtained by interchanging the numerator and the denominator. Thus, the reciprocal of \(\frac{25}{8}\) is \(\frac{8}{25}\)

Step 3: Convert the division into multiplication

Substitute division by multiplication by the reciprocal of the divisor. The problem \(-\frac{5}{16} \div \frac{25}{8}\) becomes \(-\frac{5}{16} \times \frac{8}{25}\)

Step 4: Perform the multiplication operation

To multiply the fractions, multiply the numerators together to get the numerator of the result, and multiply the denominators together to get the denominator of the result. The product of \(-\frac{5}{16} \times \frac{8}{25}\) is \(-\frac{40}{400}\)

Step 5: Simplify the result

The obtained result \(-\frac{40}{400}\) can be simplified to \(-\frac{1}{10}\) by dividing both the numerator and denominator by the greatest common factor, which is 40.

Key Concepts

Reciprocal in Fraction Division

Understanding the concept of the reciprocal is crucial for dividing fractions. A reciprocal is simply a fraction flipped upside down. For example, if we have a fraction \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \). The reciprocal changes the positions of the numerator and the denominator. This concept is essential because dividing by a fraction is the same as multiplying by its reciprocal. When you divide two fractions, you take the reciprocal of the divisor (the fraction you're dividing by) and multiply it with the dividend (the fraction you're dividing). This transformation from division to multiplication simplifies the process and is a core part of fraction division.

Simplification in Fraction Operations

Simplifying fractions is about reducing them to their lowest terms. It means finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by this number. In the problem \( -\frac{40}{400} \), we simplified the fraction by dividing both the numerator and the denominator by their GCF, which is 40. This gave us \( -\frac{1}{10} \). Simplification helps in making fractions easier to work with and understand. Always aim to bring fractions to their simplest form as the final step in any fraction operation.

Multiplication of Fractions Made Easy

When multiplying fractions, the process is straightforward: multiply the numerators to get the new numerator and multiply the denominators to get the new denominator. Using our example, \( -\frac{5}{16} \times \frac{8}{25} \) involves multiplying across the tops and bottoms. You multiply \(-5\) by \(8\) for the numerator, which results in \(-40\), and \(16\) by \(25\) for the denominator, resulting in \(400\). The result is \( -\frac{40}{400} \). Remember, multiplication and division of fractions often go hand-in-hand with simplification for clear and concise results.