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Math formula

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
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- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
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- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
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- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
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- Implicit Differentiation Tangent Line
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- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
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- Intermediate Value Theorem
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- Jump Discontinuity
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- Limit Laws
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- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
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- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
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- Population Change
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- Riemann Sum
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- Separable Equations
- Simpson's Rule
- Solid of Revolution
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- Surface Area of Revolution
- Symmetry of Functions
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- Taylor Polynomials
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- Techniques of Integration
- The Fundamental Theorem of Calculus
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- The Power Rule
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- Theorems of Continuity
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- Decision Maths
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- Altitude
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- Area and Volume
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- 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
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- Convexity in Polygons
- Coordinate Systems
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- Distance and Midpoints
- Equation of Circles
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- Figures
- Fundamentals of Geometry
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- Glide Reflections
- HL ASA and AAS
- Identity Map
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- Isometry
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- Law of Cosines
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- Plane Geometry
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- Segment Length
- Similarity
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- Squares
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- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
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- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
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- Volume of Cone
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- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- 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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- Projectiles
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- 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
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- Analyzing Graphs of Polynomials
- Angle Measure
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- 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
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
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- 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 Matrices
- 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
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- 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
- Sum of Natural Numbers
- 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
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
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- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
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- Random Variables
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- Sampling
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- Single Variable Data
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- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
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- Types of Data in Statistics
- Venn Diagrams

Say you wanted to install a wooden surface onto your bedroom floor that is in the shape of a rectangle. The length and width of your floor measure \(5\) metres by \(4\) metres long. Given these dimensions, is there a way for you to determine how many wooden panels you would need to cover your floor?

Well, since you have the length and width of your floor, you could simply use the area of a rectangle formula to identify the amount of material that you need. The area of a rectangle is given by the product of its length and width. In this case, you would need a total of \(20\) square metres worth of wooden panels to cover your floor. This is an example of a math formula.

In this article, you will look at **math formulas** and ways in which you can express them in order to use them to solve number problems.

A **formula** in mathematics is a helpful tool used to determine solutions through a given expression. By knowing the general recipe needed to solve a particular problem, you would be able to replicate the same style of working if you encounter a similar situation. This process is done through various mathematical operations.

A **math formula **is a rule in the form of a statement expressed as symbols to help solve problems easily.

Formulas consist of different quantities connected together with the equal sign. They contain variables and sometimes constants. This means that if you have the values of certain variables in a formula you can find the value of the remaining variables.

To give us a better gist of what a math formula is, let's demonstrate it with an example.

Consider the rectangle below is a plot of land owned by Mr Parker. He wants to make it into a park where children can come around from the neighbourhood to play. He wants to know the exact measurement around this land particularly, the total of all the lengths and widths. This measurement is known as the perimeter.

Dimensions of Mr Parker's land

One way to measure the perimeter of the rectangle above is to manually measure the whole plot of land. However, this can be done mathematically if some sides are known. If you know that the length is \(100\) metres and the width is \(55\) metres, you could simply use a math formula that gives you a general recipe that calculates the perimeter of a rectangle.

Carefully examining the properties of the rectangle, you will notice that the two opposite sides are equal. This means that if the length below is \(100\) metres, the length above will also be \(100\) metres. By this, you can write the formula for finding its perimeter. Let the letter \(l\) represent the length, and \(w\) represent the width:

\[ \text{Perimeter of rectangle } = l + l + w+w.\]

This can further be simplified by adding the like terms

\[ \text{Perimeter of rectangle } = 2l + 2w.\]

You can factor out \(2\) to get

\[ \text{Perimeter of rectangle } = 2(l + w).\]

Having this as the formula for finding the perimeter of a rectangle, you can go ahead to substitute numbers in it to see if it helps Mr Parker deal with his problem efficiently.

\[ \begin{align} \text{Perimeter of rectangle } &= 2(l + w) \\ &= 2(100 + 55) \\ &= 2(155) \\ &= 310 \, m. \end{align}\]

With the use of the formula, Mr Parker can simply know the perimeter of his plot of land without having to manually measure it all.

Across several mathematic fields, different formulas are being applied. To know where and how formulas can be applied, you must understand the problem you are dealing with and know which variables are significant.

As mentioned earlier, formulas are in the form of equations or identities. They consist of variables and sometimes constants. The fundamental task of writing formulas is knowing what to represent as a relevant variable.

For example, if you want to write a formula for the perimeter of a rectangle you should know that the length has a close relationship to the perimeter. You can take an example of how formulas are written.

Suppose you know that \(3\) cats eat as much food as one large dog. Write a formula to determine the volume of food you will need to feed \(27\) cats and \(10\) large dogs in terms of the number of dogs you have.

**Solution**

It is a good idea decide what you are trying to do first! You are looking to find a formula for volume given number of cats and number of dogs. So let's give these things some variables.

- \(c\) is the number of cats
- \(d\) is the number of dogs
- \(V\) is the volume of food

You are asked to find a formula for the volume of food for \(27\) cats and \(10\) dogs. What do you know? You know that \(3\) cats eat as much as one large dog. So

\[3c = 1d.\]

You want the formula for \(27\) cats and \(10\) dogs, or in other words the formula for

\[ V = 27c + 10d,\]

but you want it in terms of dogs, not dogs and cats! What to do? Well, you haven't made use of the fact that \(3c = d\). You can do a little factoring to get

\[ \begin{align} V &=27c + 10d \\ &= 9(3c) + 10d, \end{align}\]

and then substitute in \(3c = d\) to get

\[ \begin{align} V &=9(3c) + 10d \\ &= 9d+ 10d \\ &= 19d, \end{align}\]

which is a formula for the amount of food you need to feed \(27\) cats and \(10\) large dogs in terms of the number of dogs you have.

The term "most important" is a bit misleading, since it really depends on who you ask! However in this section, you will discuss some common formulas that are used across mathematics.

The area of a shape is defined by a two-dimensional region bounded by the given shape.

Concept | Formula |

Area of rectangle | Area = length \(\times\) width |

Area of parallelogram | Area = base \(\times\) height |

Area of triangle | Area = \( \dfrac{1}{2} \times\) base \(\times\) height |

Area of circle | Area = \(\pi\times \) radius\(\times\) radius |

The volume of a solid is the amount of three-dimensional space occupied by an object, container or closed surface.

Concept | Formula |

Cuboid | Volume = length \(\times\) base\(\times\) height |

Triangular prism | Volume =\( \dfrac{1}{2} \times\) length \(\times\) base\(\times\) height |

Cylinder | Volume = \(\pi\times \) radius\(\times\) radius\(\times\) height |

Compound measures are expressions that contains more than one quantity.

Concept | Formula |

Speed | \( \text{Speed } = \dfrac{ \text{ Distance}}{\text{time}}\) |

Density | \( \text{Density } = \dfrac{ \text{ Mass}}{\text{ Volume}}\) |

Pressure | \( \text{Pressure } = \dfrac{ \text{ Force}}{\text{Area}}\) |

It is useful to know how to rewrite formulas as you may be given the area of a rectangle and be asked to find its length. When you rewrite a formula the aim is to create an equation that is equivalent to the formula but with the missing variable by itself.

The fundamental rule used to do this is the golden rule of manipulating equations. It says that do unto the side of an equation what you do to the other. This means that if the manipulation requires that you add values to one side of the equation, do the same addition on the left side of the equation. Here is an example.

If the values for mass and density were given, what will be the formula for volume?

**Solution**

A formula where all the quantities mentioned are present is the formula for density.

\[\text{Density } = \dfrac{ \text{ Mass}}{\text{ Volume}}\]

To find the formula for volume, you will have to make volume the subject of the equation. This will mean that any form of manipulation on any side of the equation will require it to be replicated on the other side. To do this, you will first need to multiply both sides of the equation by volume,

\[\text{Density }\times \text{ Volume } = \dfrac{ \text{ Mass}}{\text{ Volume}} \times \text{ Volume}\]

and then cancel to get

\[\text{Density }\times \text{ Volume } = \text{ Mass}.\]

Now you can divide both sides by Density

\[\dfrac{\text{Density }\times \text{ Volume }}{\text{Density } } = \dfrac{\text{ Mass}}{\text{Density} }\]

and cancel again to get

\[\text{ Volume } = \dfrac{\text{ Mass}}{\text{Density} }.\]

Let us look at another example.

Find the length of a rectangle given the area to be \(42\, cm^2\) and its width to be \(6\, cm\).

**Solution**

First of all, you can write the formula for finding the area of a rectangle down:

\[A = lw.\]

To find the length, you will have to make it the subject of the equation. This means that you have a few manipulations to perform. What you do on one side will require that it be done on the other. To isolate length to be alone on one side of the equation, you will have to divide both sides of the equation by width

\[ \frac{A}{w} = \frac{lw}{w}\]

and then cancel to get

\[ l = \frac{A}{w}.\]

You now have a formula for finding length in this scenario. You can go ahead to find the solution to the problem by substituting into the formula:

\[ \begin{align} l &= \frac{A}{w}\\ &= \frac{42}{6} \\ &= 7.\end{align}\]

Don't forget the units! The length is \(7\, cm\).

Substitution into formulas is the process of replacing a variable with its value into a formula. In this section, the use of formulas becomes extremely evident. Given the right values of variables, the unknown variables can be found.

The whole process of substituting into formulas is replacing the letter (variables) with their values given. You will take lots of examples to see how the different types of possible situations can be approached.

Find \(z\) when \(x=7\) in the given formula

\[z = x+2.\]

**Solution**

All you have to do here is to replace \(x\) in the formula with \(7\) since the problem says \(x\) is the same as \(7\).

\[ \begin{align} z &= x+2 \\ &= 7 + 2 \\ &= 9.\end{align}\]

Here is another example for you!

Find \(l\) when \(m=5\) in the given formula

\[ l = 7m.\]

**Solution**

Here you will replace the letter \(m\) with the number \(5\) as given in the problem, then you can go ahead to find \(l\). So

\[ l = 7m.\]

The relationship between \(7\) and \(m\) here is multiplication. This whole formula can fundamentally be written as

\[l = 7 \cdot m,\]

or

\[l = 7(m).\]

Substituting \(5\) in for \(m\), you get

\[ \begin{align} l &= 7(5) \\ &= 35.\end{align}\]

- A math formula is a rule in a statement form expressed as symbols to help solve problems easily.
- Formulas consist of different quantities connected together with the equal sign.
- The fundamental rule used to rewrite formulas is the golden rule of manipulating equations which says that do unto the side of an equation what you do to the other.
- Substitution into formulas is the process of replacing a variable with its value into a formula.

A math formula is a rule in a statement form expressed as symbols to help solve problems easily.

E=mc^2

You can prove a formula through mathematical induction.

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