Problem 38 - Introductory Algebra for College Students

Question

find the multiplicative inverse of each $$\frac{1}{7}$$

Step-by-Step Solution

Answer

The multiplicative inverse of \( \frac{1}{7} \) is \( 7 \)

Step 1. Analyzing the fraction

Firstly, recognize the fraction given in the exercise as \( \frac{1}{7} \).

Step 2. Understanding the multiplicative inverse

The multiplicative inverse of a fraction is obtained by flipping the numerator and the denominator. This is based on the principle that any number multiplied by its multiplicative inverse gives the product as 1.

Step 3. Finding the multiplicative inverse

By applying the above-mentioned principle, the multiplicative inverse of \( \frac{1}{7} \) is the fraction flipped upside down, which equals \( 7 \).

Key Concepts

Fractions

Fractions are a way to express numbers that are parts of a whole. They are written with two numbers separated by a line. The number on the top is called the numerator, and the number at the bottom is the denominator. Understanding fractions helps us perform various mathematical operations like addition, subtraction, multiplication, and division. Each part has a role:

• The numerator indicates how many parts we have.
• The denominator shows the total number of equal parts the whole is divided into.

For example, in the fraction \( \frac{1}{7} \), the 1 is the numerator, and the 7 is the denominator. This fraction tells us that we have 1 part out of the total 7 parts. Fractions are crucial not just in basic math but also in advanced mathematics, engineering, and sciences.

Numerator and Denominator

In a fraction, the numerator and the denominator have specific roles and are key to understanding how fractions work.

Numerator

The numerator is the number above the fraction bar. It represents how many parts of the fraction you actually have or are considering. For example, in \( \frac{1}{7} \), the numerator is 1. This means we are looking at one part of something that is divided into seven equal parts.

Denominator

The denominator is the number below the fraction bar. It shows into how many equal parts the whole is divided. For example, the denominator in \( \frac{1}{7} \) is 7. So, the whole is divided into seven equal parts.

Importance in Fractions

• The numerator can change while the denominator remains the same, which shows different fractions of the same whole.
• Flipping the numerator and denominator gives you a reciprocal, important for operations like finding the multiplicative inverse.

Understanding these parts helps you manipulate fractions better, including turning them into mixed numbers, improper fractions, or finding reciprocals.

Basic Algebra Concepts

Algebra introduces us to concepts that simplify complex calculations using symbols and rules. In basic algebra, we often deal with equations and expressions that include fractions.One crucial algebraic operation involving fractions is finding the multiplicative inverse. The multiplicative inverse, also known as the reciprocal, is a fraction flipped upside down. This operation is based on the principle that a number multiplied by its inverse equals 1.

Why is it Useful

• Helps in solving equations where fractions need to be eliminated or simplified.
• Useful for dividing fractions; just multiply by the inverse.
• Critical for understanding proportional relationships and ratios.

For example, the multiplicative inverse of \( \frac{1}{7} \) is \( 7 \), because when you multiply them together, you get 1: \[ \frac{1}{7} \times 7 = 1 \]These concepts are foundational in algebra and provide a base for more advanced topics in mathematics.