Problem 4 - Fundamentals of Mathematics

Question

Find the reciprocal of each number. $$\frac{1}{5}$$

Step-by-Step Solution

Answer

The reciprocal of \( \frac{1}{5} \) is 5.

Step 1: Understanding the Concept of Reciprocal

A reciprocal of a number is what you multiply with your number to get the product 1. For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).

Step 2: Identify the Given Number

The number given is \( \frac{1}{5} \). We will find its reciprocal.

Step 3: Swap Numerator and Denominator

Swap the numerator and the denominator of the fraction. The reciprocal of \( \frac{1}{5} \) is \( \frac{5}{1} \).

Step 4: Simplify the Reciprocal if Possible

The fraction \( \frac{5}{1} \) simplifies to the whole number 5 because dividing by 1 does not change the number.

Key Concepts

Fractions

Understanding fractions is essential because they are a fundamental part of mathematics that you will encounter often. A fraction represents a part of a whole and is expressed as two numbers written one above the other, separated by a line. This line is known as the fraction bar. Fractions can describe quantities less than a whole, equal to a whole, or even greater than a whole when dealing with improper fractions.

• Proper Fractions: Here, the numerator (top number) is less than the denominator (bottom number), e.g., \(\frac{1}{5}\).
• Improper Fractions: In this case, the numerator is greater than or equal to the denominator, e.g., \(\frac{5}{3}\). These can also be expressed as mixed numbers, such as \(1\frac{2}{3}\).
• Equivalent Fractions: These are fractions that may look different but actually represent the same value. For example, \(\frac{1}{2}\) is equivalent to \(\frac{2}{4}\).

To solve problems involving fractions, it is crucial to understand their structure and how to manipulate them correctly. This involves understanding numerators and denominators, which we will explore next.

Numerator and Denominator

The numerator and the denominator are the two parts of a fraction. Each part plays a specific role:

• Numerator: This is the top part of the fraction and indicates how many parts of the whole you have. For example, in the fraction \(\frac{1}{5}\), 1 is the numerator, meaning you have 1 part out of a total of 5.
• Denominator: This is the bottom part of the fraction and shows into how many parts the whole is divided. In the example of \(\frac{1}{5}\), 5 is the denominator, indicating the whole is divided into 5 equal parts.

Together, the numerator and denominator provide a complete picture of what portion of a whole is being considered. It's important to note that the denominator can never be zero because division by zero is undefined in mathematics. Understanding these components helps in operations like finding reciprocals, where you switch these numbers.

Simplifying Fractions

Simplifying fractions makes them easier to work with by reducing them to their simplest form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

For instance, if you start with the fraction \(\frac{8}{12}\), the GCD of 8 and 12 is 4. By dividing both numbers by 4, you get \(\frac{2}{3}\). Therefore, \(\frac{8}{12}\) simplifies to \(\frac{2}{3}\).Simplifying makes fractions easier to read and compare. Furthermore, while working with reciprocals, as in the original problem, simplifying allows you to see straightforward results, such as realizing \(\frac{5}{1}\) is simply 5.