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# What is the most useful formulas in math?

## What is the most useful formulas in math?

Turning pages of your randomly to find the most used math formulas?

Facing difficulty in revising maths problems as your formulas seems scattered?

Not anymore. TEL Gurus bring you a complete and holistic guide of maths formulas.

Just skimp through this list.

Bingo! You are ready for your exams.

Math formulas are expressions or set of rules that help you solve the mathematics problems quickly and accurately.

From Key stages to GCSE, advance level maths concepts become easy to grasp if you know the formula.

The best way to learn the formulas is Practice, Practice and Practice.

So, let’s make your maths learning more easy and fun with these exhaustive list of maths formulas.

1. Number Theory
2. Geometry
3. Arithmetic
4. Algebra
5. Probability
6. Combinatorics (Permutation & Combination)
7. Trigonometry
8. Consumer Math Formulas
9. Statistics
10. Calculus

## 1. Number Theory

### 1.1 Number sets

Let’s discuss all the number sets with examples

## Number Sets

Number set type Definition Example
Natural Numbers All counting numbers are natural numbers N= 1,2,3,…
Prime Numbers The numbers having exactly two factors i.e. 1 and the number itself P= 2,3,5 ,7,11,…
Composite Numbers Number having more than two factors 4,6,8,10,12,14,…
Whole Numbers All natural numbers with 0 form the whole numbers W= 0,1,2,3,4,5,…
Integers All positive and negative numbers form integers. Z= ….,-4,-3,-2,-1,0,1,2,3,4,….
Rational Numbers Numbers that can be written in form of fractions. Q = 1/2, 11/10, 0.3333
Irrational Number Numbers that cannot be written as fractions. F = π, √2, ..
Real Numbers Real numbers are a set of rational and irrational numbers. R=…,−1,0,1,1.1,–√7,2,π,…
Complex Numbers Any number that can be written in the form of a + ib, where a is the real part and b is the imaginary part and i stands for √-1. C=…,−5+6i, 0, 7+9i, …

## 2D Shapes

Shape Variables Perimeter Area
Square Side =s 4s s2
Rectangle Length= l

2(I + b) lb
Circle Radius =r 2πr πr2
Triangle Side1 = a

Side2 = b

Side3 = c

s = (a+b+c)/2

s is semi perimeter

a +b +c √(s(s-a)(s-b)(s-c))
Right angled triangle Base = b

Height = h

Side 2 = a

Side 3= c

a +b +c ½ *  b*h
Parallelogram Base= b

Height = h

Other side = a

2 (a +b ) bh

## 3D shapes

Shape Variables Volume Curved Surface area / lateral surface area Total Surface Area
Cube Edge = a a3 4a2 6a2
Cuboid

height =h

lbh 2h(I + b) 2 (lb +  bh + hl)
Cylinder

Radius of the circular base =r

Height = h

πr2h 2πrh 2πr (r + h)
Cone Radius of the circular base =r

Height = h

Slant height = l

1/3  πr2h πrl πr (l + r)
Sphere Radius = r 4/3 πr3 4 πr2 4 πr2
Hemisphere Radius = r 2/3 πr3 2 πr2 3 πr2

## 2. Arithmetic Formulas

#### 2.1 BODMAS

BODMAS is an acronym that stands for brackets, Order, division, multiplication, addition and subtraction.

It tells us the order to be followed while performing arithmetic calculations.

#### 2.2 Fractions

Fractions are part of a whole.

A fraction has two parts numerator and denominator. The upper part of fraction is known as numerator and lower part as denominator.

For example: ¾  here 3 is the numerator and 4 is the denominator.

### Concepts of Fraction

Proper Fraction When numerator < denominator, the fraction is proper fraction. For example: 3/4
Improper fraction When numerator > denominator, the fraction is improper fraction. For example 8/5.
Mixed Fraction A mixed fraction has both a whole number and fractional part. For example:
Like Fractions Two fractions are said to be like fractions when they have same denominator.
Example: 2/13 and 5/13 are like fractions
Unlike Fractions Two fractions are said to be unlike fractions when they have different denominator.
Example: 2/3 and 5/8 are unlike fractions

Let a/b and c/d are two fractions where a≠b≠c≠d

#### 2.3 Progressions

Progressions are when numbers are arranged in a particular order. There are three major types of Progressions Arithmetic Progression, Geometric Progression and Harmonic Progression.

## 4. Algebra Formulas

Algebraic formulas are used to calculate expressions involving variables and constants. Here is a quick revision to all the formulas

• (a + b)2= a2 + 2ab + b2
• (a – b)2= a2 – 2ab + b2
• (a + b)(a – b) = a2– b2
• (x + a)(x + b) = x2+ x(a + b) + ab
• a2 + b2 = (a + b)2 -2ab
• (a + b + c)2= a2 + b2 + c2 + 2ab + 2bc + 2ca
• (a – b – c)2= a2 + b2 + c2 – 2ab + 2bc – 2ca
• (a + b)3 = (a + b)3 = a3 + b3 + 3ab(a + b)
• (a – b)3 = a3 – b3 – 3ab(a – b)
• (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
• (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4

Here are some formulas involving exponents having same bases with different powers or different bases with same powers.

• am. an = am+n
• am / an = am-n
• (am)n = amn
• (ab)m = am. bn
• a0 = 1
• (a)-m = 1/am

For any quadratic equation ax2 + bx + c = 0

Where x is the variable and a,b,c  are constants, the roots can be calculated by using the formula

b2 – 4ac is also called determinant (D) and helps in finding the nature of the roots.

• D > 0, then the quadratic equation has two distinct real roots.
• D = 0, then the quadratic equation has two equal real roots.
• D < 0, then the quadratic equation has two imaginary roots.

## 5. Probability Formulas

The likelihood of an event happening is known as probability (p).

The probability of not happening an event is 1 – p.

Probability formula

P(A) = Number of favourable Outcomes / Total number of favourable outcomes

Here P(A) is the probability of an event A.

Or, it is also written as

P(A) = n (A) /n (S)

n(A) stands for number of favourable outcomes

n(S) stands for total number of events in the sample space

Let’s discuss some probability formulas for two events say, A and B.

## Most useful Probability formulas

Probability Range 0 ≤ P(A) ≤ 1
Additional Rule P(A∪B) = P(A) + P(B) – P(A∩B)
Complementary Addition P(A’) + P(A) = 1
Independent events P(A∩B) = P(A) * P(B)
Disjoint events P(A∩B) = 0
Bayes’ Formula P(A | B) = P(B | A) * P(A) / P(B)
Conditional Probability P(A | B) = P(A∩B) / P(B)

Let’s revise with an example:

 Q1. Calculate the probability of getting an even number on rolling a dice.  Solution: Sample space (S) = { 1,2,3,4,5,6} n(S) = 6 Let A be event of getting even number (favourable outcome) = {2,4,6} n(A) = 3 Probability of getting an even number = n(A)/n(S) = 3/6 = ½ So, the probability of getting an even number on rolling a dice is ½.

## 6. Combinatorics

Let there be n things, they can be arranged in n factorial Ways.

n factorial is written as n! and can be calculated as n (n-1) (n-2)…3*2 *1.

Conventionally 0! is considered as 1.

#### Permutation and combination

The primary difference between permutation and combination is the order.

If the order of things matter, it’s a permutation and if it doesn’t it’s a combination.

Permutations of n things taken r at a time can be determined by:

P(n,r) = n! / (n−r)!

Combinations of n things taken r at a time can be determined by:

C(n,r) = n! / (n−r)! r!   or,

C(n,r) = P(n,r)  / r!

Let’s understand this with the help of an example:

 Permutation: Picking a Captain and Vice captain from a group of 10 sportspersons.P(10,2) = 10! / 8! = 10 * 9 = 90 Combination: Picking a team of 4 students from a class of 10 students.C (10,4) = 10! / 4!6! = 10 * 9 * 8 *7 / (4* 3 *2 * 1)              = 210

## 7. Trigonometry

This branch of mathematics deals with the study of triangles and relationships between its side lengths and angles.

### 7.1 Pythagoras Theorem

In a right angled triangle,

(Hypotenuse)2 = (Base)2 + (perpendicular)2

There are six main trigonometric ratios sine, cosine, tangents, cosecant, secant and cotangent.

### 7.2 Basic trigonometric ratio formulas

Taking right angled triangle as base with one of the angle as θ.

sin θ = Opposite Side/Hypotenuse  or, O/H

cos θ = Adjacent Side/Hypotenuse or, A/H

tan θ = Opposite Side/Adjacent Side or, O/A

sec θ = Hypotenuse/Adjacent Side or, H/A

cosec θ = Hypotenuse/Opposite Side or, H/O

cot θ = Adjacent Side/Opposite Side Or, A/H

### 7.3 Reciprocal Identities

In a right angled triangle with an angle θ

• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ
• sec θ = 1/cos θ
• cosec θ = 1/sin θ
• cot θ = 1/tan θ

Also, tan θ = sin θ /cos θ

cot θ = cos θ/ sin θ

cos2 θ + sin2 θ = 1

### 7.4 Co-function Identities

The co-function or periodic identities can also be represented in degrees as:

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = cosec x
• cosec(90°−x) = sec x

## Trigonometric Table

Angle degrees (Radians) 0° (0°) 30° (π/6) 45° (π/4) 60° (π/3) 90° (π/2)
Sine 0 1/2 1/√2 √3/2 1
Cosine 1 √3/2 1/√2 1/2 0
Tangent 0 1/√3 1 √3

Let’s learn with an example:

 Example: Solve (cos 30° + sin 30°) – (cos 60° + sin 60°)? Solution: From the trigonometric table we can get the values of all the ratios cos 30° =  √3/2 sin 30° = 1/2 cos 60° =  1/2 sin 60° = √3/2 Substituting the values in (cos 30° + sin 30°) – (cos 60° + sin 60°) We get,  (√3/2 + 1/2) – (1/2 + √3/2) = 0

## 8. Consumer Math Formulas

### 8.1 Percentage

A fraction with denominator 100 is considered as percentage. The symbol “%” is used instead of over 100.

• Converting any number into a percentage

Multiply by 100%.

• Converting any percentage into a number

Remove the percentage sign and divide by 100

• Percent increase

% increase = (increase/ original value)* 100%

• Percent Decrease

% increase = (Decrease / original value)* 100%

 Q1. In the exam, John obtained 406 marks out of 500. Calculate the percentage of marks obtained by John.  Solution: % marks = (marks obtained/ total marks) * 100% = (406/500) * 100% = 81.6% John obtained 81.6% marks in the exam.

### 8.2 Profit/Loss

If an item is sold above the cost price, it is said to have been sold at a profit.

Profit = Selling Price – Cost Price

Profit% = (Profit/Cost Price ) * 100%

If an item is sold below the cost price, it is said to be sold at a loss.

Loss = Cost Price – Selling Price

Loss% = (Loss/Cost Price) * 100%

Some other formulas:

Discount = List Price * Discount rate

Discount rate = discount / list price * 100%

Sale Price = List Price – Discount

Sales tax = Price of item * tax rate

### 8.3 Interest Calculation

#### Simple Interest

Simple Interest = Principal * rate of Interest * Time

Amount = Principal + Simple Inertest

#### Compound Interest  (CI)

CI = Interest on the principal + CI at regular intervals

CI = P ( 1 + r/100)nt – P

Here,

P is principal

r is rate of interest

n is the number of times interest gets compounded annually

t is the time frame

If the principal amount is compounded annually n becomes 1.

The formula becomes:

CI = P ( 1 + r/100)t – P

Amount = P (1 + r/100)t

## 9. Statistics

When it comes to analysing and dealing with data and numbers, Statistics is used. It’s a branch of mathematics that performs the study of interpretation, collection, analysis, organization and presentation of data.

Let’s have look at most useful formulas in Statistics.

Let’s understand this with help of an example:

 Example 1: Here are the marks obtained by the students in exams = {140,150,160,150,170,150,180}. Find the mode.   Solution: Since, 150 is the only value that has the maximum frequency i.e. 3. Mode = {150} So, mode in the list is 150.

## 10.     Calculus

Calculus is the branch of mathematics that majorly involves the study of “rate of change” that includes convergence of infinite sequences or within a defined limit.

Differential Calculus and Integral Calculus are two major branches of Calculus. While differentiation splits area into small parts, integration does the reverse.

Let’s have a look at the common formulas related to both.

### 10.2  Integration Formulas

• ∫xn.dx = xn+1/(n+1) + C
• ∫1.dx = x +  C
• ∫ex.dx = ex + C
• ∫(1/x).dx = log |x| + C
• ∫ax.dx = (ax/ln a) + C
• ∫cos x.dx = sin x + C
• ∫sin x.dx = -cos x + C
• ∫sec2 x dx = tan x + C
• ∫cosec2 x dx = -cot x + C
• ∫sec tan x dx = sec x + C
• ∫cosec x. cot x dx = -cosec x + C
• ∫a dx = ax + C
• ∫ln x dx = x.ln x – x + C
• ∫tan x dx = ln |sec x| + C
• ∫tan2 x dx = tan x –x +C
• ∫sec x tan x dx = sec x + C
• ∫cot x. dx = ln |sin x| + C
• ∫sec x dx = ln |sec x + tan x| + C
• ∫cosec x dx = ln |cosec x – cot x | + C

Where C is known as integration constant

So, get these formulas by heart and start preparing for your exams.

Good Luck!

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