Properties of Fractional Division

Let’s dive into the properties of fractional division. Understanding how to divide fractions involves grasping a few fundamental concepts and properties.

Here are the key properties and steps:

1. Reciprocal Property

Definition: The reciprocal (or multiplicative inverse) of a fraction is obtained by flipping the numerator and the denominator.

Example: The reciprocal of a/b is b/a​.

2. Division as Multiplication by the Reciprocal

Rule: Dividing by a fraction is the same as multiplying by its reciprocal.

Example: a/b÷c/d=a/b×d/c

Explanation: Instead of dividing by c/d​, we multiply by its reciprocal, d/c​.

3. Multiplying Fractions

Rule: To multiply two fractions, multiply their numerators together and their denominators together.

Example: a/b×d/c=(a×d)/(b×c)

4. Simplification

Rule: Simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Example (axd)/(bxc) can be simplified if axd and bxc have a common factor.

Steps to Divide Fractions

Let’s walk through the steps of dividing fractions using an example:

Example: 3/4÷2/5

Step 1: Find the reciprocal of the divisor (the second fraction).

2/5 becomes 5/2

Step 2: Multiply the dividend (the first fraction) by the reciprocal of the divisor.

3/4×5/2

Step 3: Multiply the numerators together and the denominators together.

(3×5)/(4×2)=15/8​

Step 4: Simplify the fraction if possible. In this case, 15/8 is already in its simplest form.

Why This Works

The reason we multiply by the reciprocal when dividing fractions can be understood through the concept of equivalent operations. When you divide by a fraction, you are essentially determining how many times that fraction fits into another fraction. Multiplying by the reciprocal achieves the same result mathematically, which simplifies the operation.

Properties in Action

1. Reciprocal Property: Ensures we can convert division into multiplication.
2. Multiplicative Property of Fractions: Allows us to handle multiplication of fractions straightforwardly.
3. Simplification: Helps us reduce the fraction to its simplest form for easier interpretation and further calculations.

Example Problems

1. Problem: 7/3÷5/6
2. ​Solution: 7/3×6/5=(7×6)/(3×5)=42/15=14/5
3. Problem: 9/2÷4/3
4. Solution: 9/2×3/4=(9×3)/(2×4)=27/8​

Understanding these properties and steps helps in mastering the division of fractions, making it a systematic and logical process.

Note:

(1) When a fractional number is divided by 1, the quotient is the fractional number itself.

Example: 3/4 ÷1/1

= 3/4 x 1/1 = (3 x 1)/(4 x 1) = 3/4

(2) When any non zero fractional number is divided by itself, then the quotient is 1.

(21/12)÷(21/12)=(21/12)x(12/21)=(21×12)/(12×21)= 1

(3) The reciprocal of zero or multiplicative inverse does not exist. So a fractional number cannot be divided by zero(0).

0÷5/2 = 0 x 2/5 = 0

Zero will never change when yoy multiply or divide any number by it.