Problem 30 - Algebra A Combined Function

Question

Find each reciprocal. \(-\frac{6}{13}\)

Step-by-Step Solution

Answer

The reciprocal of \(-\frac{6}{13}\) is \(-\frac{13}{6}\).

Step 1: Understand the Definition of Reciprocal

The reciprocal of a number is what you multiply that number by to get 1. For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).

Step 2: Identify the Numerator and Denominator

In the fraction \(-\frac{6}{13}\), the numerator is -6, and the denominator is 13.

Step 3: Swap the Numerator and Denominator

To find the reciprocal, swap the numerator and the denominator of the given fraction. This means the fraction \(-\frac{6}{13}\) becomes \(\frac{13}{-6}\).

Step 4: Simplify the Reciprocal (Optional)

The reciprocal \( \frac{13}{-6} \) is already in its simplest form as a fraction with a negative sign. It can also be written as \(-\frac{13}{6}\) for clarity, indicating the whole fraction is negative.

Key Concepts

Fractions

Fractions are a way to express parts of a whole. They are represented by two numbers, one on top of the other, separated by a line. This format can be thought of as a division problem.
For example, the fraction \( \frac{6}{13} \) signifies six parts out of a total of thirteen parts. The number above the line, 6, is called the numerator, and the number below the line, 13, is called the denominator.
Fractions can describe numbers less than one or more than one, and they come in various types, such as proper fractions (where the numerator is smaller than the denominator), improper fractions (where the numerator is larger), and mixed numbers (whole numbers combined with a fraction).

• Proper fractions: \( \frac{3}{4} \)
• Improper fractions: \( \frac{5}{3} \)
• Mixed numbers: 2 \( \frac{1}{2} \)

Understanding fractions helps in operations like addition, subtraction, multiplication, and division, as they often appear in real-world contexts, such as measuring ingredients in a recipe or dividing portions.

Numerators and Denominators

The terms numerator and denominator are fundamental concepts in fractions. The numerator is the top number, indicating how many parts are considered. The denominator, the bottom number, tells you how many equal parts make up the whole. These terms are crucial when working with fractions because they determine the size of the fraction.
For example, in the fraction \( \frac{6}{13} \), the numerator is 6, meaning six parts are considered out of 13. The denominator is 13, indicating that the whole is divided into 13 parts.

When calculating with fractions, altering the numerator and denominator changes the fraction's value. Fractions are often simplified by dividing both the numerator and the denominator by their greatest common divisor, but this doesn't affect the fraction's overall value. You can think of the numerator as a counter and the denominator as a divider.

• Numerator example: In \( \frac{11}{4} \), 11 is the numerator.
• Denominator example: In \( \frac{11}{4} \), 4 is the denominator.

This distinction is vital when performing operations such as finding the reciprocal, where you swap these numbers.

Negative Numbers

Negative numbers extend the idea of numbers to include values less than zero. They are an essential part of the number line, found to the left of zero. Negative numbers are used to express losses, depths below sea level, and temperatures below freezing, among other things.
In mathematics, having a negative number means that the direction in a certain context is inverse.

• Example of negative numbers on a number line: -3, -2, -1, 0, 1, 2, 3
• Using negative numbers to represent temperature: -5°C means 5 degrees below zero.

When working with fractions involving negative numbers, like \(-\frac{6}{13}\), the negative sign indicates that the fraction represents a value less than zero. This can mean any of the following:

• The whole fraction as a negative: -\(\frac{6}{13}\)
• Or the numerator being negative: \(\frac{-6}{13}\).

In reciprocal operations, it is crucial to understand where the negative sign belongs to ensure accurate calculations and interpretations. While recasting a fraction like \(-\frac{6}{13}\) to its reciprocal \(-\frac{13}{6}\), the negative sign can shift to either the numerator or denominator, owing to properties of fractions and negatives.