Problem 21 - Fundamentals of Mathematics

Question

For the following problems, find the reciprocal of each number. $$\frac{4}{5}$$

Step-by-Step Solution

Answer

The reciprocal of \( \frac{4}{5} \) is \( \frac{5}{4} \).

Step 1: Identify the number

The given number is \( \frac{4}{5} \). We need to find the reciprocal of this fraction. A reciprocal is simply swapping the numerator and the denominator of the fraction.

Step 2: Swap numerator and denominator

To find the reciprocal, swap the numerator and the denominator. This means the numerator 4 becomes the denominator and the denominator 5 becomes the numerator. So, the reciprocal of \( \frac{4}{5} \) is \( \frac{5}{4} \).

Key Concepts

Numerator

In mathematics, the numerator of a fraction is the number above the fraction bar. It represents how many parts of a whole are being considered. For example, in the fraction \( \frac{4}{5} \), the number 4 is the numerator.

Understanding the role of the numerator can simplify learning about fractions and reciprocals.

• The numerator indicates the quantity we are working with.
• In a reciprocal, the numerator becomes the new denominator.

In the given example, to find the reciprocal of \( \frac{4}{5} \), the numerator 4 swaps places with the denominator 5. This process results in a new fraction, \( \frac{5}{4} \), where 5 is now the numerator.

Recognizing that the numerator can change positions helps when dealing with more complex mathematical operations involving fractions.

Denominator

The denominator is the number located below the fraction bar. It tells us the total number of equal parts that make up a whole. In \( \frac{4}{5} \), the number 5 is the denominator.

Here's why understanding the denominator is crucial:

• It provides the framework to gauge the size of each fractional part.
• In reciprocals, the denominator becomes the numerator.

When calculating the reciprocal of a fraction like \( \frac{4}{5} \), you swap the numerator and denominator. Thus, the denominator of 5 turns into the numerator, creating the fraction \( \frac{5}{4} \).

The change of roles between numerator and denominator when creating reciprocals is fundamental to mastering fraction manipulation.

Fraction

Fractions are a way to represent parts of a whole. They consist of a numerator and a denominator separated by a fraction bar. In our example, \( \frac{4}{5} \) represents 4 parts of a whole divided into 5 equal parts.

Fractions are used to describe quantities that are not whole numbers.

• They are essential for understanding ratios and proportions.
• Fractions allow for the division of whole numbers into smaller parts.

A critical aspect of fractions is the concept of the reciprocal. For a fraction \( \frac{a}{b} \), swapping the numerator with the denominator gives you its reciprocal, \( \frac{b}{a} \).

Understanding the structure and function of fractions, and how to manipulate them through operations like finding reciprocals, is fundamental in math, helping you solve various problems more easily.