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.
Simple Fractions - Basics and Definition
A simple fraction consists of two numbers separated by a fraction bar:
- Numerator (top number) - indicates how many parts are taken
- Denominator (bottom number) - shows how many equal parts the whole is divided into
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
Reciprocals - What They Are and How to Form Them
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.
Interesting Property of Reciprocals
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 Role of Reciprocals in Mathematics
Dividing Fractions
The primary use of reciprocals is in dividing fractions.
Dividing by a fraction = Multiplying by its reciprocal
Example: \( \large \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6} \)
Solving Equations
Reciprocals simplify equations involving fractions.
To eliminate a fraction, multiply both sides of the equation by its reciprocal.
Example:
Practical Applications of Reciprocals
In cooking
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).
In finance
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.
In physics
Electrical resistance (R) is the reciprocal of conductance (G): \( \large R = \frac{1}{G} \).
Practice - Test Yourself!
Task 1: Find the reciprocals
- \( \large \frac{3}{5} \)
- \( \large \frac{7}{2} \)
- \( \large \frac{4}{9} \)
Task 2: Divide fractions using reciprocals
- \( \large \frac{1}{3} \div \frac{2}{5} \)
- \( \large \frac{5}{6} \div \frac{10}{3} \)
- \( \large \frac{4}{7} \div \frac{2}{7} \)
Task 3: Apply reciprocals in equations
- \( \large \frac{3}{4}x = 15 \)
- \( \large \frac{2}{5}y = 8 \)
- \( \large \frac{5}{9}z = 20 \)
Summary
- A reciprocal is formed by swapping the numerator and denominator.
- The product of a fraction and its reciprocal is always 1.
- Dividing by a fraction equals multiplying by its reciprocal.
- Reciprocals are used in mathematics, finance, physics, and daily life.
- A fraction equal to 0 has no reciprocal.
Reciprocals are key to many mathematical operations, especially division!