Number System

Percentage

Profit and Loss

Simple Interest and Compound Interest

Ratio and Proportion

Average

Time and Work

Pipes and Cisterns

Time, Speed and Distance

Boats and Streams

Partnership

Mixture and Alligation

Permutation and Combination

Probability

Algebra

Geometry

Mensuration

Data Interpretation

Concept

Formula

Comparing two numbers in terms of percentage

$X \text{ is what percentage of } Y = \frac{X}{Y} \times 100$

$X \text{ is what percentage more/less than } Y = \frac{|X - Y|}{Y} \times 100$

To find the percentage or fraction

$X \text{ % of } Y = \frac{XY}{100}$

Successive Change (If a value is increased or decreased successively by $R_1$% and $R_2$%)

Net % change = $R_1 + R_2 + \frac {R_1R_2}{100}$%

Where $R_1$ and $R_2$ are taken as positive for increase and negative for decrease.

Concept

Formula

Relating Selling Price (SP), Cost Price (CP), and Profit, Loss

$S.P. = C.P. + \text{Profit}$

$S.P. = C.P. - \text{Loss}$

To find Profit or Loss Percentage

Profit/Loss % = $\frac {\text{Profit or Loss}}{CP} \times 100$%

To find the discount Percentage

Discount % = $\frac {\text{Discount}}{MP} \times 100$% where MP is marked Price

To find the Markup Percentage

Mark up % = $\frac {\text{Mark up value}}{CP} \times 100$%

Relating Selling Price and Cost Price with Profit%

$S.P. = \frac{C.P. \times 100 + \text{Profit%} \times 100}{100}$

$C.P. = \frac{S.P. \times 100}{100+ \text{Profit%}}$

Relating Selling Price and Cost Price with Loss%

$S.P. = \frac{C.P. \times 100 - \text{Loss%} \times 100}{100}$

$C.P. = \frac{S.P. \times 100}{100-\text{Loss%}}$

Concept

Formula

Simple Interest

For Principal ($P$), Rate of Interest ($R$), Time ($T$):

$S.I. = P \times R \times T / 100$

Compound Interest (compounded annually)

$A = P(1 + \dfrac{R}{100})^n$

where $n$ = time in years.
CI = A - P

Compound Interest (compounded half-yearly)

$A = P \left(1 + \dfrac{R}{2 \times 100}\right)^{2T}$

Difference Between C.I. and S.I. (2 years)

$C.I. - S.I. = P \left( \dfrac{R}{100} \right)^2$

Difference Between C.I. and S.I. (3 years)

$C.I. - S.I. = P \left( \dfrac{R}{100} \right)^2 \left( 3 + \dfrac{R}{100} \right)$

Concept

Formula

Relating speed, distance, and time

Distance = Speed $\times$ Time

Average Speed

$\text{Average Speed} = \dfrac{\text{Total Distance}}{\text{Total Time}}$

Average speed when the distance covered in each stage of a journey is the same, but speeds are different

$ \text{Average Speed} = \frac{2 s_1 s_2}{s_1 + s_2} $

Average speed when the time taken in each stage is the same but speeds differ

$ \text{Average Speed} = \frac{s_1 + s_2}{2} $

Trains

Time for a train to cross a pole/person

$T = \dfrac{l}{s}$; Where: $l =$ Length of the train, $s =$ Speed of the train

Time for a train to cross a platform/tunnel

$T = \dfrac{l + d}{s}$; Where: $l =$ Length of the train, $d =$ Length of platform/tunnel, $s =$ Speed of the train

Time for trains to cross each other (same direction)

$T = \dfrac{l_1 + l_2}{s_1 - s_2}$; Where: $l_1, l_2 =$ Lengths of Train 1 and Train 2; $s_1, s_2 =$ Speeds of Train 1 and Train 2

Time for trains to cross each other (opposite direction)

$T = \dfrac{l_1 + l_2}{s_1 + s_2}$; Where: $l_1, l_2 =$ Lengths of Train 1 and Train 2; $s_1, s_2 =$ Speeds of Train 1 and Train 2

Boats and Stream

Distance (Downstream: same direction)

$D = (x + y) \times t$ km where x is speed of boat in still water, y is the speed of stream and t is the time.

Distance (Upstream: opposite direction)

$D = (x - y) \times t$ km where x is speed of boat in still water, y is the speed of stream and t is the time.

Both are moving in the same direction

Both are moving in opposite directions

Time of first meeting anywhere on the track

$ t_{\text{first}} = \frac{D}{|a-b|} $

$ t_{\text{first}} = \frac{D}{a+b} $

Time to meet again at the starting point

$ t_{\text{start}} = \text{LCM}\left(\frac{D}{a}, \frac{D}{b}\right) $

$ t_{\text{start}} = \text{LCM}\left(\frac{D}{a}, \frac{D}{b}\right) $

Number of distinct meeting points:

$ \text{Meeting points} = |x - y| $ Where $x:y$ is the simplified ratio of speeds.

$ \text{Meeting points} = |x + y| $ Where $x:y$ is the simplified ratio of speeds.

Concept

Formula

Individual work

If A can do work in x days: A’s rate = $\frac {1}{x}$

Combined Work

If A and B work together: Combined rate = 1/x + 1/y;

Time = 1 / (1/x + 1/y) = (x × y) / (x + y)

Work rate

Work = Rate × Time

Divisibility Rules

Divisible by 2: last digit even

Divisible by 3: sum of digits divisible by 3

Divisible by 4: last two digits are divisible by 4

Divisible by 5: last digit 0 or 5

Divisible by 6: divisible by 2 and 3

Divisible by 8: last three digits are divisible by 8

Divisible by 9: sum of digits is divisible by 9

Divisible by 11: difference of alternate digit sums is divisible by 11

HCF and LCM

For two numbers:

Product of two numbers = HCF × LCM

Basic Remainder Theorem

Dividend = Divisor × Quotient + Remainder

Sum Formulas

Sum of first n natural numbers = $ \frac{n(n+1)}{2}$

Sum of first n even numbers = $n(n+1)$

Sum of first n odd numbers = $n^2$

Sum of squares of first $n$ natural numbers = $\frac{n(n+1)(2n+1)}{6} $

Sum of cubes of first $n$ natural numbers = $\left(\frac{n(n+1)}{2}\right)^2 $

Basic Identities

$(a+b)^2 = a^2 + b^2 + 2ab$

$(a-b)^2 = a^2 + b^2 - 2ab$

$a^2-b^2 = (a-b)(a+b)$

$(a+b)^3 = a^3+b^3+3ab(a+b)$

$(a-b)^3 = a^3-b^3-3ab(a-b)$

$a^3+b^3 = (a+b)(a^2-ab+b^2)$

$a^3-b^3 = (a-b)(a^2+ab+b^2)$

Quadratic Equation

For $ax^2+bx+c=0$

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Logarithm Basics

Definition of Logarithm:
$ \log_b a = x ;;\Longleftrightarrow;; b^x = a $

Log of 1: $ \log_b 1 = 0 \quad (b>0, ; b \neq 1) $

Log of the base itself: $ \log_b b = 1 $

Log of a product: $ \log_b (mn) = \log_b m + \log_b n $

Log of a quotient: $ \log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n $

Log of a power: $ \log_b (m^n) = n \cdot \log_b m $

Change of base formula: $ \log_b a = \frac{\log_k a}{\log_k b} $

Base switch rule: $ \log_a b = \frac{1}{\log_b a} $

Area of 2D figures

Triangle: $\frac{1}{2} \times \text{base} \times \text{height}$

Heron’s formula: $\sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$

Rectangle: length × breadth

Square: side²

Parallelogram: base × height

Circle: $\pi r^2$

Trapezium: $\frac{1}{2}(a+b)h$ where $a,b$ are parallel sides.

Rhombus: $\frac{1}{2}d_1d_2$ where $d_1, d_2$ are diagonals.

Cube

Volume = $a^3$

Total Surface Area = $6a^2$

Diagonal = $a\sqrt{3}$

Cuboid

Volume = $lbh$

Total Surface Area = $2(lb+bh+hl)$

Diagonal = $\sqrt{l^2+b^2+h^2$

Cylinder

Volume = $\pi r^2 h$

Curved Surface Area = $2\pi rh$

Total Surface Area = $2\pi r(r+h)$

Cone

Volume = $\frac{1}{3}\pi r^2 h$

Curved Surface Area = $\pi rl$

Total Surface Area = $\pi r^2+\pi rl$

Sphere

Volume = $\frac{4}{3}\pi r^3$

Surface Area = $4\pi r^2$

Hemisphere

Volume = $\frac{2}{3}\pi r^3$

Surface Area = $3\pi r^2$

Curved Surface Area = $2\pi r^2$