Simple Fractions and Reciprocals

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:

\( \large \frac{1}{2}, \frac{3}{4}, \frac{2}{3}, \frac{5}{8} \)

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.

If \( \large \frac{a}{b} \) is a fraction, then \( \large \frac{b}{a} \) is its reciprocal.

Example: The reciprocal of \( \large \frac{3}{4} \) is \( \large \frac{4}{3} \).

Example 1:

\( \large \frac{2}{5} \rightarrow \frac{5}{2} \)

The reciprocal of \( \large \frac{2}{5} \) is \( \large \frac{5}{2} = 2\frac{1}{2} \).

Example 2:

\( \large \frac{7}{3} \rightarrow \frac{3}{7} \)

The reciprocal of \( \large \frac{7}{3} = 2\frac{1}{3} \) is \( \large \frac{3}{7} \).

Example 3:

\( \large \frac{1}{4} \rightarrow \frac{4}{1} = 4 \)

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:

\( \huge \frac{a}{b} \times \frac{b}{a} = \frac{a \times b}{b \times a} = \frac{ab}{ab} = 1 \)

Example 1:

\( \large \frac{2}{3} \times \frac{3}{2} = \frac{2 \times 3}{3 \times 2} = \frac{6}{6} = 1 \)

Example 2:

\( \large \frac{5}{8} \times \frac{8}{5} = \frac{5 \times 8}{8 \times 5} = \frac{40}{40} = 1 \)

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

\( \large \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \)

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:

\( \large \frac{2}{3}x = 10 \)
\( \large \frac{3}{2} \times \frac{2}{3}x = \frac{3}{2} \times 10 \)
\( \large x = 15 \)

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

Reciprocals are key to many mathematical operations, especially division!

Suma de Fracciones Resta de Fracciones Multiplicación de Fracciones División de Fracciones

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