FRACTIONS | Study

Fractions: Understanding the Basics

Modules:

Introduction of fractions (Picture Based, Verbal)

Decimals And Fractions

Simple fractions and simplification of fractions

GCD (Greatest Common Divisor)

Order Of Fractions (Increasing And Decreasing)

LCM (Least Common Multiple)

Product Of Numbers = Product Of LCM and GCD Of The Numbers

Addition of fractions

LCM (Least Common Multiple)

Rewriting fractions with a common denominator

Subtraction Of Fractions:

Multiplication Of Fractions:

Division Of Fractions

Fractions And Decimals

Introduction to Fractions

Fraction is the representation of the considered number of equal parts out of the total equal parts.

Example: ½. ⅔, 3/2 etc.

We use fractions at different situations such as:

Case 1:

When a single whole item is divided into more than one equal part.

Case 2:

Two or more whole items are divided into more than two equal parts.

This is the special case of addition or subtraction of case 1.

Etymology of Fractions:

The word “fraction” comes from Latin, where “fractus” means “broken.” It’s like breaking something into smaller pieces.

Representation of A Fraction:

A Fraction has the following three parts:

Parts of a Fraction:

A fraction consists of three parts:

Numerator: The upper part of the fraction, representing the selected or shaded sections.

Denominator: The lower part, indicating the total number of parts into which the whole is divided.

Fraction Bar Or Division Line: Fraction Bar Or Division Line is a bar that separates the numerator and denominator.

Example: If we have the fraction 3/4, then 3 is the numerator, and 4 is the denominator.

Rational number: 

Rational number is a number used to represent a fraction.

In other words, Fraction is a numerical representation of the considered number of equal parts out of the total equal parts.

A Rational Number is represented as a numerator parts out of the denominator parts.

Examples:

Half (1/2):

Imagine cutting an apple into two equal parts. Each part represents a half of the apple.

One Third (1/3):

Divide a chocolate bar into three equal pieces. Each piece is a third of the whole chocolate.

Quarter (1/4):

Cut a sandwich into four equal parts. Each part is a quarter of the sandwich.

Types of Fractions:

Proper Fraction: 

The numerator is smaller than the denominator (e.g., 2/5).

Improper Fraction: 

The numerator is equal to or greater than the denominator (e.g., 7/4).

Mixed Fraction: 

Combines a whole number and a proper fraction (e.g., 1 3/4).

Like Fractions: 

Have the same denominators (e.g., 3/5 and 2/5).

Unlike Fractions: 

Have different denominators (e.g., 1/3 and 2/5).

Equivalent Fractions: 

Represent the same portion of a whole (e.g., 1/2 and 2/4).

Unit Fraction:

A unit fraction has a numerator of 1 (e.g., 1/3, 1/5).

It represents one equal part out of the whole.

Visualizing Fractions on a Number Line:

Place fractions on a number line to understand their relative positions.

For example, 1/2 lies exactly halfway between 0 and 1.

Operations with Fractions:

Addition Of Fractions: 

Adding Fractions: A Step-by-Step Guide

1. Check the Denominators:
   • First, make sure the denominators (the bottom numbers) are the same for both fractions.
   • If they already have the same denominator, you’re dealing with like fractions.
   • Example: Adding 1/4 + 2/4 (both have a denominator of 4).
2. Add the Numerators:
   • The numerator is the number on top of the fraction.
   • Simply add the numerators together, just like you would with regular whole numbers.
   • Example:
     • 1/4 + 2/4 = 3/4 (3 = 1 + 2)
3. Place the New Numerator Over the Common Denominator:
   • Take the sum of the numerators and place it on top of the fraction.
   • The denominator remains the same (don’t add the denominators together).
   • Example:
     • 1/4 + 2/4 = 3/4 (numerator = 3, denominator = 4)
4. Simplify the Fraction (if Possible):
   • If the numerator and denominator have a common factor, divide both by that factor to simplify the fraction.
   • Example:
     • 9/8 (from 3/8 + 2/8 + 4/8) can be simplified to 1 1/8.

Remember, adding fractions is like sharing and combining parts of a whole. Practice these steps, and soon you’ll be a pro at adding all types of fractions! 

Create step by step Proces of addition of Fractions.

Multiplying Fractions: A Step-by-Step Guide

1. Multiply the Numerators:
   • Start by multiplying the numerators (the top numbers) of the fractions.
   • Example: Multiply 3/4 by 2/5.
     • Numerator product: (3 \times 2 = 6).
2. Multiply the Denominators:
   • Next, multiply the denominators (the bottom numbers) of the fractions.
   • Example: Multiply 3/4 by 2/5.
     • Denominator product: (4 \times 5 = 20).
3. Combine the Results:
   • Place the numerator product over the denominator product to form the new fraction.
   • Example: Multiply 3/4 by 2/5.
     • Result: \(\frac{6}{20}\)

1. Simplify (if Needed):
   • To simplify the fraction, find the greatest common factor (GCF) of the numerator and denominator.
   • Divide both the numerator and denominator by the GCF.
   • Example: Simplify \(\frac{6}{20}\)
     • GCF of 6 and 20 is 2.
     • Simplified result \(\frac{3}{10}\)

Remember, multiplying fractions is like finding the area of a part of a whole. Practice these steps, and soon you’ll be a pro at multiplying fractions!

Division Of Fractions: 

Dividing fractions

Dividing fractions is the same as multiplying by the reciprocal (inverse).

Dividing Fractions: A Step-by-Step Guide

1. Check the Denominators:
   • First, take the reciprocal (flip) of the second fraction (the divisor).
   • Example: If you’re dividing (\frac{3}{4}) by (\frac{2}{5}), the reciprocal of (\frac{2}{5}) is (\frac{5}{2}).
2. Multiply the Numerators:
   • Multiply the numerators (the top numbers) of both fractions.
   • Example:
     • Numerator product: (3 \times 5 = 15).
3. Multiply the Denominators:
   • Multiply the denominators (the bottom numbers) of both fractions.
   • Example:
     • Denominator product: (4 \times 2 = 8).
4. Form the Resultant Fraction:
   • Place the numerator product over the denominator product.
   • Example:
     • Result: (\frac{15}{8}).
5. Simplify (if Needed):
   • If possible, simplify the fraction by finding the greatest common factor (GCF) of the numerator and denominator.
   • Example:
     • Simplified result: (\frac{15}{8}) can be expressed as (\frac{1}{\frac{8}{15}}).

Remember, dividing fractions is like sharing parts of a whole. Practice these steps, and soon you’ll be a pro at dividing fractions! 

Properties of Fractions:

Fractions share properties similar to real numbers:

Commutative and Associative Properties hold true for fractional addition and multiplication.

The identity element for fractional addition is 0, and for multiplication, it’s 1.