Simple fractions and reciprocals are fundamental concepts in mathematics. A reciprocal is formed by swapping the numerator and denominator. Below, you’ll find clear explanations and practical applications.
A simple fraction consists of two numbers separated by a fraction bar:
For example, the fraction \( \large\frac{3}{4} \) means the whole is divided into 4 equal parts, and 3 parts are taken.
Examples of simple fractions:
Simple fractions are used to represent parts of a whole in mathematics, cooking, construction, and many other fields.
Uses of simple fractions:
In the kitchen
\( \large\frac{3}{4} \) cup of flour, \( \large\frac{1}{2} \) teaspoon of salt
In construction
A board with a thickness of \( \large\frac{3}{4} \) inch
In music
Notes like \( \large\frac{1}{4} \), \( \large\frac{1}{8} \) to determine duration
Definition:
A reciprocal is a fraction created by swapping the numerator and denominator of the original fraction.
Example: The reciprocal of \( \large \frac{3}{4} \) is \( \large \frac{4}{3} \).
Example 1:
The reciprocal of \( \large \frac{2}{5} \) is \( \large \frac{5}{2} = 2\frac{1}{2} \).
Example 2:
The reciprocal of \( \large \frac{7}{3} = 2\frac{1}{3} \) is \( \large \frac{3}{7} \).
Example 3:
The reciprocal of \( \large \frac{1}{4} \) is \( \large \frac{4}{1} = 4 \) (a whole number).
Important!
The reciprocal of a fraction \( \large \frac{a}{b} \) exists only if \( \large a \neq 0 \). You cannot form a reciprocal for \( \large \frac{0}{b} \), as it would result in division by zero.
The product of a fraction and its reciprocal is always 1:
Example 1:
Example 2:
This property is crucial in algebra and equation solving, as it allows you to eliminate a fraction by multiplying both sides of an equation by its reciprocal.
The primary use of reciprocals is in dividing fractions.
Dividing by a fraction = Multiplying by its reciprocal
Example: \( \large \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} \)
Reciprocals simplify equations involving fractions.
To eliminate a fraction, multiply both sides of the equation by its reciprocal.
Example:
If a recipe serves 4 and you’re cooking for 6, multiply the ingredients by \( \large\frac{6}{4} = \frac{3}{2} \) (or 1.5).
If the EUR/USD exchange rate is 1.10 USD per EUR, the reverse rate (USD/EUR) is \( \large\frac{1}{1.10} \) EUR per USD.
Electrical resistance (R) is the reciprocal of conductance (G): \( \large R = \frac{1}{G} \).
Task 1: Find the reciprocals
Task 2: Divide fractions using reciprocals
Task 3: Apply reciprocals in equations
Reciprocals are key to many mathematical operations, especially division!