Problem 57 - Beginning and Intermediate Algebra

Question

Find each reciprocal or multiplicative inverse. $$ -\frac{3}{11} $$

Step-by-Step Solution

Answer

The reciprocal is \(-\frac{11}{3}\).

Step 1: Understanding the Reciprocal

The reciprocal (or multiplicative inverse) of a number is another number which, when multiplied with the original number, results in 1. The reciprocal of a fraction is found by swapping the numerator and the denominator.

Step 2: Applying the Concept

To find the reciprocal of \(-\frac{3}{11}\), swap the numerator and the denominator, reversing the sign as needed. Thus, the reciprocal is \(-\frac{11}{3}\).

Step 3: Verify the Solution

Multiply the original fraction \(-\frac{3}{11}\) with its reciprocal \(-\frac{11}{3}\): \(-\frac{3}{11} \times -\frac{11}{3} = \frac{33}{33} = 1\). This confirms that the reciprocal is correct.

Key Concepts

Multiplicative Inverse

The concept of a multiplicative inverse, also known as the reciprocal, is essential in mathematics, especially when dealing with fractions and algebraic expressions. When you multiply a number by its multiplicative inverse, the result is always 1. For example, if you have a fraction like \(-\frac{3}{11}\), its multiplicative inverse is found by swapping the numerator and the denominator, resulting in \(-\frac{11}{3}\). The negative sign is retained, as the inverse should mirror the original sign to maintain the product as positive. Understanding the multiplicative inverse helps in solving equations where you need to isolate a variable or simplify complex fractions. Always remember: The multiplicative inverse of a number is pivotal because it enables us to "undo" multiplication, similar to how subtraction undoes addition.

Fractions

Fractions represent a part of a whole and consist of two parts: the numerator and the denominator. The fraction \(-\frac{3}{11}\) illustrates the concept of a fraction, where \(-3\) is the numerator, and \(11\) is the denominator. Fractions can be positive or negative. The sign indicates the direction or nature of the quantity. When working with fractions, it's possible to perform various operations like addition, subtraction, multiplication, and division. Each operation might require different considerations. For instance, finding the reciprocal (as in this exercise) involves flipping the numerator and the denominator while preserving the original sign. Fractions are a fundamental concept in mathematics that provide a way to express and operate with parts of a whole in a precise manner.

Numerator and Denominator

The numerator and denominator are crucial in understanding and working with fractions. In any fraction \(\frac{a}{b}\), "\(a\)" is the numerator, and "\(b\)" is the denominator.

• The numerator represents the number of parts considered.
• The denominator represents the total number of equal parts in the whole.
• Swapping these values creates the reciprocal.

In the fraction \(-\frac{3}{11}\), swapping the numerator and denominator transforms it into \(-\frac{11}{3}\). This switch is fundamental when finding reciprocals. Understanding the roles and positions of the numerator and denominator helps you navigate through calculations in fractions, making such mathematical operations more intuitive.

Algebra

Algebra is an area of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It builds upon concepts like fractions, reciprocals, and operations with numbers. In algebraic expressions, reciprocals are often used to simplify equations and solve for variables. When you encounter an algebraic fraction, finding its reciprocal follows the same basic principle: switch the numerator and the denominator and retain any sign. This action allows you to multiply equations by the reciprocal, effectively isolating the variable of interest. For example, if we start with trying to solve an equation and we have a term like \(\frac{2x}{15}\), its reciprocal \(\frac{15}{2x}\) might be used to simplify or eliminate the fraction. Thus, reciprocals serve as a powerful tool in algebra to manipulate and solve equations efficiently.