Problem 41 - Introductory Algebra for College Students

Question

find the multiplicative inverse of each $$-\frac{2}{5}$$

Step-by-Step Solution

Answer

The multiplicative inverse of -2/5 is -5/2.

Step 1: Identify the numerator and denominator of the fraction

The given fraction is -2/5. Here, the numerator is -2 and the denominator is 5.

Step 2: Swap the numerator and the denominator

Write the fraction as a division -2 ÷ 5. To find the multiplicative inverse (or the reciprocal), swap the places of the numerator and the denominator. Therefore, the division becomes 5 ÷ -2.

Step 3: Write the result as a fraction

The swapped fraction 5 ÷ -2 is the multiplicative inverse of -2/5. It can be written as -5/2.

Key Concepts

Understanding Fraction Operations

When it comes to navigating the world of fractions, mastering fraction operations is critical. A fraction consists of two parts: the numerator, which is the top number and represents the parts you have, and the denominator, which is the bottom number and represents the total number of parts.

Operations with fractions often involve four key procedures: addition, subtraction, multiplication, and division. However, before you can perform these operations, you need to ensure the fractions are on common ground, which typically means having the same denominator. Multiplication and division of fractions come with their own set of rules. When multiplying, you simply multiply the numerators together and then the denominators, whereas division involves flipping the second fraction and then multiplying.

Effectively Simplifying Fractions

In simplifying or reducing fractions, the goal is to make the fraction as simple as possible. This is done by finding a common factor of both the numerator and the denominator and dividing them both by it. Simplification is a fundamental step before undertaking any fraction operations.

The Reciprocal of a Fraction

A reciprocal, simply put, is what you multiply a number by to get 1. In the realm of fractions, finding the reciprocal includes flipping the numerator and the denominator. For instance, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). It's essential to remember that the reciprocal of a whole number 'a' is \(\frac{1}{a}\), and the reciprocal of \(\frac{1}{a}\) is 'a'.

To find the reciprocal of a negative fraction, like \(-\frac{2}{5}\), you apply the same process: flip the numerator and the denominator. Therefore, the reciprocal is \(-\frac{5}{2}\). It's crucial to keep the negative sign; it can be placed either on the numerator or the denominator, or even in front of the fraction as a whole, but not in more than one place at a time.

Understanding Reciprocals in Equations

The concept of a reciprocal is especially important when solving equations involving fractions. When you multiply a fraction by its reciprocal, you always get 1, which is a useful property for solving algebraic equations.

Algebraic Concepts

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols; it's a language that describes patterns, relationships, and changes. One foundational concept within algebra is the multiplicative inverse, or reciprocal as discussed. Another important algebraic concept is the variable, a symbol that represents a number which we may not know yet. It's often represented by letters like 'x' or 'y'.

In algebra, we also encounter equations, which are statements of equality that contain variables. An equation like \(x \times \frac{2}{5} = 1\) can be solved by multiplying both sides by the reciprocal of \(\frac{2}{5}\), which is \(-\frac{5}{2}\).

Applying Algebraic Concepts in Real Life

These concepts are not just abstract; they are used to solve real-world problems. Whether it's in engineering, economics, or everyday problem-solving, understanding how to manipulate algebraic expressions and equations is vital.