Problem 83 - Prealgebra and Introductory Algebra

Question

Perform the indicated operation. $$-\frac{3}{8} \div \frac{7}{8}$$

Step-by-Step Solution

Answer

After performing the indicated operation, the result is \(-\frac{3}{7}\).

Step 1: Understanding the problem

The division indicated in this exercise is \(-\frac{3}{8} \div \frac{7}{8}\). This is the same as multiplying \(-\frac{3}{8}\) by the reciprocal of \(\frac{7}{8}\).

Step 2: Finding the reciprocal

The reciprocal of a fraction is obtained by inverting (swapping) the numerator and denominator. Thus, the reciprocal of \(\frac{7}{8}\) is \(\frac{8}{7}\).

Step 3: Performing the multiplication

Multiply \(-\frac{3}{8}\) by the reciprocal \(\frac{8}{7}\). Apply the rule for multiplication of fractions, which is to multiply the numerators together (to get a new numerator) and the denominators together (to get a new denominator), to achieve the result \(-\frac{3 \cdot 8}{8 \cdot 7}\).

Step 4: Simplifying the result

Now, simplify by canceling the common factors. Here, 8 is a common factor in the numerator and denominator, therefore the result is \(-\frac{3}{7}\).

Key Concepts

Reciprocal of a Fraction

The reciprocal of a fraction is a fundamental concept in fraction division. It refers to flipping the numerator and the denominator of a given fraction. For example, if we have a fraction \( \frac{a}{b} \), its reciprocal will be \( \frac{b}{a} \). This means the top number becomes the bottom number and vice versa.

Reciprocals are essential when dividing fractions. Instead of directly dividing, we multiply by the reciprocal. It simplifies the process and converts division into multiplication, which is generally easier to handle. In our original problem, the reciprocal of \( \frac{7}{8} \) is \( \frac{8}{7} \).

• To find a reciprocal: just swap the numerator and the denominator.
• For whole numbers, like 5, treat them as fractions (\( \frac{5}{1} \)) before finding the reciprocal.

Multiplication of Fractions

Multiplying fractions is a straightforward process once you grasp the steps. Unlike addition or subtraction, there's no need for common denominators here. When multiplying fractions, you multiply the numerators together and the denominators together.

For example, multiplying \( \frac{a}{b} \) and \( \frac{c}{d} \) is done by multiplying the numerators (\( a \times c \)) and the denominators (\( b \times d \)) to get \( \frac{ac}{bd} \).

• Write the fractions side by side.
• Multiply the top numbers (numerators).
• Multiply the bottom numbers (denominators).
• The result forms the numerator and denominator of your answer.

In our example of \( -\frac{3}{8} \times \frac{8}{7} \), we multiply -3 by 8 for the new numerator and 8 by 7 for the new denominator, resulting in \( -\frac{24}{56} \).

Multiplying helps us simplify later on if there are common factors in the numerators and denominators.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest or simplest form. This makes them not only easier to understand but also to work with in further calculations.

To simplify a fraction, divide both the numerator and the denominator by their greatest common factor (GCF). This helps you find the simplification that reduces the fraction while keeping the value the same.

• Identify the common factors of the numerator and denominator.
• Divide both by the largest factor they share.
• Ensure the fraction is in its simplest form with no common factors left other than 1.

In the solution, after multiplying \( -\frac{3}{8} \times \frac{8}{7} \) resulting in \( -\frac{24}{56} \), we simplify by recognizing that 8 is a common factor. Cancel the 8s to get \( -\frac{3}{7} \), the simplest form of the fraction.

Simplifying fractions helps achieve neatness and precision in your final result.