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Fractions

- Calculus
- Absolute Maxima and Minima
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Candidate Test
- Combining Differentiation Rules
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Disk Method
- Divergence Test
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Improper Integrals
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrating Even and Odd Functions
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Differential Equation
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- One-Sided Limits
- Optimization Problems
- P Series
- Particular Solutions to Differential Equations
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Tangent Lines
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
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- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
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- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal´s Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Comparing Data
- Conditional Probability
- Correlation
- Cumulative Frequency
- Data Interpretation
- Discrete Random Variable
- Distributions
- Events (Probability)
- Frequency Polygons
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Testing
- Large Data Set
- Linear Interpolation
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Random Variables
- Sampling
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Measures
- Tree Diagram
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

Do you know that the United Kingdom is divided into 4 major regions, this means that each region is a fraction of the UK?

In this article, we introduce fractions definition, types of fractions as well as arithmetic operations on fractions.

A fraction is defined as a part or portion out of a whole. This whole can be anything including even a fraction but not zero.

The idea of fractions is strongly linked with the division operation because effectively, fractions are formed when you try to share something.

Imagine a box of pizza delivered to your doorstep and you want to share it with your friend, you would have to share it (divide) into 2 parts. Then each one of you will be having half of the pizza, which is expressed as,

Now if you intend to take the lion's share by splitting it into 4 parts and giving just 1 out of the four pieces to your friend, then you have given 1 part out of 4 parts which are expressed as,

.

However, you have actually taken 3 out of for parts, which is expressed as,

Below shows a figure for a better illustration,

A fraction is composed basically of three components, **the numerator, the over sign, and the denominator.**

The numerator is the number or expression on top or above the over sign.

For example, in the fraction , 2 is the numerator.

The over sign ( ) separates the numerator from the denominator as seen in the fraction .

The denominator is the number or expression underneath or below the over sign.

For example, in the fraction , 3 is the denominator.

There are three types of fractions which are,

- Proper fraction
- Improper fraction

- Mixed fraction

A proper fraction is a fraction where the numerator is less than the denominator.

In the fraction , 3 being the numerator is less than the denominator 5.

Further proper fractions are etc.

An improper fraction is a fraction where the numerator is greater than the denominator.

In the fraction , 7 being the numerator is greater than the denominator 6.

Further improper fractions are etc.

A mixed fraction consists of a whole number and a proper fraction.

The mixed fraction consists of

1 as the whole number,

2 as the numerator of the proper fraction in the mixed fraction, and

3 as the denominator of the proper fraction in the mixed fraction.

Other examples include, etc.

Improper fractions can be converted to mixed fractions as well as mixed fractions can be converted to improper fractions.

We recall that an improper fraction has its numerator greater than its denominator. This means that the numerator is divisible by the denominator but with a remainder. To convert an improper fraction to a mixed fraction we follow these steps,

- Divide the numerator by the denominator.
- Put the whole number on the left-hand side.
- Put your remainder as your new numerator on the right-hand side of the whole number.
- Retain your former denominator.

Convert to a mixed fraction.

**Solution**

The quotient from the division of 7 by 3 is 2, and the remainder is 1.

Thus, 2 is your whole number that stays on the left-hand side and 1 is the remainder that becomes your new numerator and 3 is your denominator that remains unchanged.

Thus,

In order to convert a mixed fraction to an improper fraction follow the following steps,

- Find the product between the denominator and the whole number.
- Find the sum of the numerator and your product.
- Your sum becomes the new numerator.
- Your denominator remains the same.

Convert to an improper fraction.

**Solution**

First, we find the product between the denominator which is 3, and the whole number which is 2, thus,

.

Next, we find the sum between the numerator which is 1 and the product found in the first step which is 6, thus

Now your sum is 7, the new numerator is thus 7, and the denominator being 3 remains the same. Therefore,

.

Equivalent fractions are **fractions that represent the same value, even though they look different.**

They differ by a multiplication factor or by the expression of fraction types.

Having 3 parts of cake from the total of 6 parts is the same as having half the cake.

Equivalent fractions can differ by a multiplying factor when one fraction is a multiple of the other fraction. Note that the multiplying factor (multiplier) affects both the numerator and the denominator.

The fractions and are equivalent but differ by a multiplying factor, 2.

In fact,

Similarly, we have

Therefore, the best way of knowing if two fractions are equivalent, but differ by a multiplying factor, is by simplifying these fractions to the simplest indivisible forms.

Equivalent fractions can differ by fraction type, so this happens only between a mixed fraction and an improper fraction.

Remember a mixed fraction can be converted to an improper fraction and vice versa.

Apparently, the two fractions are the same but are only distinct because of their fraction types.

is the same as , the only difference is that is a mixed fraction while is an improper fraction.

For fractions to be added or subtracted, we follow the following steps,

- If whole numbers are already present add/subtract and keep them aside (left-hand side) because we add these whole numbers to our result afterward.
- We must find the lowest common denominator (LCD) between the denominators of the fractions.
- Use the lowest common denominator as the new general denominator.
- Divide the new denominator by the former denominator and multiply the result by the respective numerator.
- Compute our result. Note that when carrying out subtraction the first (left) value may be less than our second (right) value. In this case, subtract 1 from the whole number which we kept aside. That 1 becomes the value of your denominator once we put it into the expression. Add the denominator value to the former value to get a number larger than our second value before subtracting.
- We may simplify if there is a common factor between the new numerator and the new denominator.
- If any whole numbers were present initially, we add them to our result.

Note that the above steps should also be followed when adding and subtracting as well as operations involving multiplication and division of mixed fractions.

Calculate the following

**Solution**

Step 1.

We see that there is no whole number.

Step 2.

The LCD between 5 and 3 is 15, thus 15 is the new general denominator.

Step 3.

We divide the new denominator by the former denominator and multiply the result by the respective numerator. Therefore we have

Now, we compute our results to get,

Calculate

**Solution**

Step 1.

We solve the whole numbers first,

We keep our result on the left-hand side.

Step 2.

The LCD between 2 and 4 is 4. Thus 4 is the new general denominator.

Step 3.

We divide the new denominator by the former denominator and multiply our result by the respective numerator. Therefore we have,

Now, we compute the result to get,

Note that 2 is less than 3, so take 1 from the whole number 3, so that your whole number is now 2. The 1 you have taken when put into the expression becomes equal to your denominator value which is 4. Add it to what is already there. Therefore, we have

Fractions like other numbers can be multiplied, as well as divided.

Fractions are multiplied by multiplying the numerators together and the denominators together.

We can always look out for factors that can be divided through numerators and denominators of any fraction. We just have to ensure that the division is between a numerator and a denominator. We should not divide a numerator and another numerator likewise a denominator and denominator.

Calculate

**Solution**

Multiply both numerators and denominators together to get,

Always confirm that no factor can divide both the numerator and denominator before giving your final answer.

Calculate

**Solution**

This can be solved using two methods.

Method 1.

Just multiply directly the numerators and denominators.

A common mistake of students is to leave the fraction in this form. If you do this, you would be wrong because the fraction is not in the simplified form.

The next step to take is to divide both the numerator and denominator by their greatest common divisor (GCD). The GCD between 24 and 36 is 12. Hence,

Moreover, if your not sure of how to find the GCD, start by dividing with the smallest number from 2, 3,..

Method 2.

You can directly simplify the fractions while multiplying them as seen below.

We notice that this method is direct and faster.

In order to divide fractions, we apply the multiplication principles to the multiplication of the first fraction with the inverse of the second fraction.

The inverse of a fraction can be found while interchanging the positions of the denominator and the numerator.

Once this fraction is inversed, the division sign is replaced with the multiplication sign.

Calculate

**Solution**

Step 1.

Inverse the fraction at the right of the division sign and change the division sign to multiplication sign to get,

Step 2.

Multiply the numerators and denominators.

Calculate,

**Solution**

Step 1.

Inverse the fraction at the right of the division sign and change the division sign to multiplication sign, to get,

Step 2.

We now simplify to get,

We shall be looking at word problems that are fraction problems.

What is the sum of one-third of a pie and half of a pie?

**Solution**One-third of a pie is and half of a pie is .

The sum means the addition of both fractions. Thus their sum is given by

Their LCD is 6, thus, we have

If one-fifth of a man's income is used on accommodation and half of the rest is used on groceries, what fraction of his income is used on groceries?

**Solution**

Step 1.

First, the fraction of the rest of the man's income after deducing that of the accommodation is

Step 2.

The rest of the man's income is four-fifth. We are told that he used half of the rest for groceries. Therefore, the fraction of the groceries is

.

- A fraction is defined as a part or portion out of a whole. This whole can be anything including even a fraction but not zero.
- A fraction is composed basically of three components: the numerator, the over sign and the denominator.
There are three types of fractions which are proper, improper and mixed fractions.

Improper fractions can be changed to mixed fractions as well as mixed fractions can be converted to improper fractions.

Equivalent fractions are two or more fractions that are the same.

A fraction is defined as a part or portion out of a whole. An example is 2/3.

The three types of fractions are proper, improper and mixed fraction.

More about Fractions

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