Core Mastery

Arithmetic & Commercial Math

Percentages & Successive Changes

Percentage

Net Effective Percentage

For two successive changes of \(a\%\) and \(b\%\), the total change is:

\[ \text{Net} = a + b + \frac{ab}{100} \]
Percentage

Consumption Balancing

If price increases by \(r\%\), reduction in consumption to keep expenditure same:

\[ \text{Reduction} = \left[ \frac{r}{100 + r} \right] \times 100\% \]
Population

Population after n Years

For constant growth rate \(R\%\) per annum:

\[ P_n = P_0 \left(1 + \frac{R}{100}\right)^n \]

Profit, Loss & Discount

Commercial

CP, MP & Discount Relation

Direct bridge between Cost Price and Marked Price using Profit and Discount:

\[ \frac{\text{CP}}{\text{MP}} = \frac{100 - D\%}{100 + P\%} \]
Vendor Tricks

Dishonest Dealer Gain

Profit when a dealer uses false weight instead of true weight:

\[ \text{Gain}\% = \frac{\text{Error}}{\text{True Value} - \text{Error}} \times 100 \]
Offers

Effective Discount (Buy X Get Y)

Calculating real savings on "Buy X Get Y Free" promotions:

\[ \text{Discount}\% = \frac{Y}{X + Y} \times 100 \]

Time, Speed, Distance & Work

Motion

Average Speed (Standard)

For equal distances covered at speeds \(x\) and \(y\):

\[ V_{avg} = \frac{2xy}{x + y} \]
Boats

Hydrodynamic Vectors

Calculating Boat (\(x\)) and Stream (\(y\)) speeds from Up/Downstream speeds:

\[ x = \frac{D + U}{2}, \quad y = \frac{D - U}{2} \]
Work

Man-Days Chain Rule

Relationship between Men, Days, Hours and Work efficiency:

\[ \frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2} \]

Ages & Ratios

Ratio

Proportional Scaling

Mean Proportional: \( \sqrt{ab} \)
Third Proportional: \( b^2/a \)

Ages

Ages Ratio Constant Difference

In age problems, the absolute difference between two people's ages remains constant over time.

Fundamental Logic

Number System & Simplification

Divisibility & Unit Digits

Units

Unit Digit Cyclicity

Patterns of last digits for powers of numbers:

\[ 2 \to 2, 4, 8, 6 \text{ (Cycle 4)} \] \[ 3 \to 3, 9, 7, 1 \text{ (Cycle 4)} \] \[ 7 \to 7, 9, 3, 1, \quad 8 \to 8, 4, 2, 6 \]
Divisibility

Composite Divisibility

6: 2 & 3 | 12: 3 & 4 | 15: 3 & 5 | 72: 8 & 9 | 88: 8 & 11

Remainders

Remainder Theorem

When \( f(x) \) is divided by \( (x-a) \), the remainder is \( f(a) \).

HCF, LCM & Factorials

Identity

Product Identity

\[ n_1 \times n_2 = \text{HCF} \times \text{LCM} \]
Factorials

Legendre's Formula

Highest power of prime \(p\) in \(n!\):

\[ E_p(n!) = \lfloor \frac{n}{p} \rfloor + \lfloor \frac{n}{p^2} \rfloor + \dots \]
Abstract Reasoning

Algebra & Progressions

Equations & Identities

Quadratic

Vieta's Formulas

For \( ax^2 + bx + c = 0 \):

\[ \alpha + \beta = -b/a, \quad \alpha\beta = c/a \]
Identity

Trinomial Sum Cube

If \( a+b+c = 0 \), then:

\[ a^3 + b^3 + c^3 = 3abc \]
Logarithm

Change of Base

\[ \log_a b = \frac{\log_c b}{\log_c a} \]

Arithmetic & Geometric Progressions

AP

AP: n-th Term & Sum

\[ T_n = a + (n-1)d \] \[ S_n = \frac{n}{2}[2a + (n-1)d] \]
GP

GP: n-th Term & Sum

\[ T_n = ar^{n-1} \] \[ S_n = \frac{a(r^n - 1)}{r - 1} \quad (r \neq 1) \]
Infinite

Infinite GP Sum

Condition: \( |r| < 1 \)

\[ S_\infty = \frac{a}{1 - r} \]
Spatial Analysis

Geometry & Mensuration

2D Shapes & Properties

Triangle

Heron's Formula

Where \( s = (a+b+c)/2 \):

\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
Circle

Sector & Arc

Arc Length: \( \frac{\theta}{360} 2\pi r \)
Sector Area: \( \frac{\theta}{360} \pi r^2 \)

Polygon

Internal Angles & Diagonals

Sum of Angles: \( (n-2) \times 180^\circ \)
Diagonals: \( \frac{n(n-3)}{2} \)

3D Solids & Frustums

3D

Sphere & Hemisphere

Sphere Vol: \( \frac{4}{3}\pi r^3 \)
Hemisphere TSA: \( 3\pi r^2 \)

Frustum

Cone Frustum Volume

\[ V = \frac{1}{3}\pi h (R^2 + r^2 + Rr) \]
Advanced Patterns

Modern Math & Probability

Permutations & Combinations

P&C

Arrangements vs Selections

\[ ^n P_r = \frac{n!}{(n-r)!}, \quad ^n C_r = \frac{n!}{r!(n-r)!} \]
Circular

Circular Permutations

Arranging \(n\) distinct objects in a circle:

\[ \text{Count} = (n-1)! \]

Probability Foundations

Prob

Classical Probability

\[ P(A) = \frac{n(E)}{n(S)} \]
Prob

Complementary Event

Probability of an event NOT happening:

\[ P(A') = 1 - P(A) \]
Prob

Bayes' Theorem (Simplified)

Probability of A given B has occurred:

\[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \]

Clocks & Calendars

Clock

Angular Separation

Angle between Hour and Minute hands at H:M:

\[ \theta = |30H - 5.5M| \]
Calendar

Odd Days Concept

Ordinary Year: 1 Odd Day | Leap Year: 2 Odd Days

Data Analysis

Statistics & Logical Reasoning

Measures of Central Tendency

Stats

Empirical Relationship

Relation between Mode, Median, and Mean:

\[ \text{Mode} = 3\text{Median} - 2\text{Mean} \]
Stats

Standard Deviation (\(\sigma\))

\[ \sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}} \]

Logical Matrix

Logic

Ranking Identity

\[ \text{Total} = (\text{Left} + \text{Right}) - 1 \]
Logic

Handshake Formula

Number of handshakes among \(n\) people:

\[ \text{Handshakes} = \frac{n(n-1)}{2} \]