Problem 138 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain how to divide fractions and give an example.

Step-by-Step Solution

Answer

To divide fractions, you multiply the first fraction by the reciprocal of the second fraction. For example, to divide 2/3 by 3/4, you multiply 2/3 by 4/3 to get 8/9.

Step 1: Understanding the Concept of Reciprocal

The reciprocal of a number is 1 divided by that number. For a fraction, the reciprocal is obtained by switching the numerator and the denominator.

Step 2: Explaining the Division of Fractions

To divide fractions, you multiply the first fraction by the reciprocal of the second fraction.

Step 3: Providing an Example

Let's take the fractions 2/3 and 3/4 as an example. To divide 2/3 by 3/4, you multiply 2/3 by the reciprocal of 3/4, which is 4/3. The multiplication is then as follows: (2/3) * (4/3) = 8/9.

Key Concepts

Reciprocal

In the realm of fractions, understanding the concept of the reciprocal is essential. It can be thought of as flipping a fraction.
For any given fraction, the reciprocal is achieved by swapping the numerator and the denominator. So, if your fraction is \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
This concept becomes especially useful when dividing fractions, as multiplying by a reciprocal is the key to solving these types of problems.

• To find the reciprocal of a whole number, imagine it as a fraction with a denominator of 1. For instance, the reciprocal of 5 is \( \frac{1}{5} \).
• For fractions like \( \frac{2}{3} \), the reciprocal would be \( \frac{3}{2} \).

Numerator

The numerator in a fraction is the number above the line. It indicates how many parts of a whole or set are being considered.
Think of it as the "counter" in the fraction, signifying how much of the pie you have. For example, in the fraction \( \frac{2}{3} \), 2 is the numerator.
It tells us we have two parts of something that is divided into three equal sections.

• The numerator can be any whole number.
• When we change the fraction (by finding the reciprocal, multiplying, etc.), the numerator often changes too.

Understanding the numerator is vital for performing operations like multiplying and dividing fractions.

Denominator

Moving on to the denominator, this is the number found below the line in a fraction. It tells us into how many equal parts the whole is divided.
Essentially, it's the name of the fractional parts we are dealing with. In the fraction \( \frac{2}{3} \), the denominator is 3, meaning the whole is divided into three equal parts.

• A denominator cannot be zero, as division by zero is undefined.
• When finding a reciprocal, the original denominator turns into the new numerator.

By understanding the denominator, you gain insight into how the parts of the fraction work together.

Multiplying Fractions

When multiplying fractions, things become straightforward. You simply multiply across the numerators and across the denominators. So, for fractions like \( \frac{a}{b} \) and \( \frac{c}{d} \), you multiply to get \( \frac{a \, \times \, c}{b \, \times \, d} \).
This operation is crucial when dividing fractions, since you multiply by the reciprocal of the divisor.

• The process is simple: find the reciprocals if needed, and multiply both the numerators and the denominators.
• After multiplying, simplify the fraction if possible, to its simplest form.

This method ensures accurate results, and mastering it means being proficient in both multiplication and division of fractions.