2.1.1 Essential Definitions & The Division Algorithm

When an integer $a$ is divided by a positive integer $b$, it yields a quotient $q$ and a remainder $r$. This relationship is governed by the fundamental division algorithm:

$$a = b \cdot q + r$$

Where $0 \le r < b$. The remainder $r$ must always be a non-negative integer strictly less than the divisor $b$.

2.1.2 Fundamental Divisibility Rules

To solve complex quantitative problems efficiently under time constraints, you must determine divisibility dynamically without resorting to long division:

• Divisibility by 3: The sum of the digits must be completely divisible by 3.
• Divisibility by 4: The number formed by the last two digits must be divisible by 4.
• Divisibility by 6: The number must simultaneously satisfy the divisibility rules for both 2 (even number) and 3.
• Divisibility by 8: The number formed by the last three digits must be divisible by 8.
• Divisibility by 9: The sum of the digits must be completely divisible by 9.
• Divisibility by 11: The absolute difference between the sum of the digits at odd places and the sum of the digits at even places must be either 0 or a multiple of 11.

2.1.3 Advanced Remainder Theorems & Power Cycles

When evaluating large exponential remainders of the form $\frac{a^n}{b}$, look for patterns where the base $a$ is close to a multiple of the divisor $b$.

$$\text{If } a \equiv 1 \pmod{b}, \text{ then } a^n \equiv 1^n \equiv 1 \pmod{b}$$

$$\text{If } a \equiv -1 \pmod{b} \equiv (b-1) \pmod{b}, \text{ then } a^n \equiv (-1)^n \pmod{b}$$

Worked Example:

• Problem: Find the remainder when $7^{40}$ is divided by 6.
• Solution: Express the base in terms of the divisor: $7 = 6 + 1$. Therefore, $7 \equiv 1 \pmod{6}$. Substitute this into the power: $1^{40} = 1$. The remainder is 1.

2.2.1 Core Formula Framework

The average or arithmetic mean represents the central value of a finite, discrete data distribution.

$$\text{Average } (\bar{x}) = \frac{\text{Sum of All Individual Observations } (\sum x)}{\text{Total Number of Observations } (n)}$$

$$\text{Sum of Observations } (\sum x) = \text{Average } (\bar{x}) \times n$$

2.2.2 Consecutive Number Properties

For any evenly spaced sequence (arithmetic progression), the average is simply the exact middle term. If the number of terms is even, the average is the arithmetic mean of the two central terms.

$$\text{Average of an AP} = \frac{\text{First Term } (a) + \text{Last Term } (l)}{2}$$

2.2.3 Advanced Weighted Average & Data Alteration

When combining two distinct groups with different counts ($n_1, n_2$) and distinct averages ($A_1, A_2$), you must apply the weighted average formula rather than averaging the averages:

$$A_{\text{weighted}} = \frac{n_1 \cdot A_1 + n_2 \cdot A_2}{n_1 + n_2}$$

Worked Example:

• Problem: The average score of 15 business students is 80, and the average score of 10 IT students is 90. What is the combined average score?
• Solution:
• $$A_{\text{weighted}} = \frac{(15 \times 80) + (10 \times 90)}{15 + 10} = \frac{1200 + 900}{25} = \frac{2100}{25} = \mathbf{84}$$

2.2.1 Core Formula Framework

The average or arithmetic mean represents the central value of a finite, discrete data distribution.

$$\text{Average } (\bar{x}) = \frac{\text{Sum of All Individual Observations } (\sum x)}{\text{Total Number of Observations } (n)}$$

$$\text{Sum of Observations } (\sum x) = \text{Average } (\bar{x}) \times n$$

2.2.2 Consecutive Number Properties

For any evenly spaced sequence (arithmetic progression), the average is simply the exact middle term. If the number of terms is even, the average is the arithmetic mean of the two central terms.

$$\text{Average of an AP} = \frac{\text{First Term } (a) + \text{Last Term } (l)}{2}$$

2.2.3 Advanced Weighted Average & Data Alteration

When combining two distinct groups with different counts ($n_1, n_2$) and distinct averages ($A_1, A_2$), you must apply the weighted average formula rather than averaging the averages:

$$A_{\text{weighted}} = \frac{n_1 \cdot A_1 + n_2 \cdot A_2}{n_1 + n_2}$$

Worked Example:

• Problem: The average score of 15 business students is 80, and the average score of 10 IT students is 90. What is the combined average score?
• Solution:
• $$A_{\text{weighted}} = \frac{(15 \times 80) + (10 \times 90)}{15 + 10} = \frac{1200 + 900}{25} = \frac{2100}{25} = \mathbf{84}$$

2.3.1 Mathematical Definitions

A ratio is a comparative mathematical tool that relates two quantities of identical units to show scale. A proportion is a structural equation stating that two distinct ratios are equal.

$$\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } a:b :: c:d \implies a \cdot d = b \cdot c$$

Where $a$ and $d$ are the extremes, and $b$ and $c$ are the means.

2.3.2 Structural Ratio Balancing

CMAT questions frequently require you to combine independent, disjoint ratios sharing a common variable.

Worked Example:

• Problem: If $A:B = 2:3$ and $B:C = 4:5$, find the unified continuous ratio $A:B:C$.
• Solution: The bridging variable is $B$. Multiply the first ratio by 4 (the value of $B$ in the second ratio) and the second ratio by 3 (the value of $B$ in the first ratio).
• $$A:B = 2 \times 4 : 3 \times 4 = 8:12$$
• $$B:C = 4 \times 3 : 5 \times 3 = 12:15$$
• $$\text{Unified Ratio } A:B:C = \mathbf{8:12:15}$$

2.3.3 Variations (Direct and Inverse)

• Direct Variation: $Y = k \cdot X$ (As $X$ scales up, $Y$ scales up proportionally).
• Inverse Variation: $Y = \frac{k}{X}$ (As $X$ scales up, $Y$ drops proportionally).

2.4.1 Fundamental Calculations

Percentages convert fractions into parts per hundred.

$$\text{Percentage Value} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100$$

$$\text{Percentage Change } (\%) = \left( \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \right) \times 100$$

2.4.2 Sequential / Successive Percentage Changes

When an asset value or population changes by $x\%$ and then successively by $y\%$, the cumulative net change is calculated using an algebraic formula that accounts for compounding. Do not simply add the values.

$$\text{Net Percentage Change} = x + y + \frac{x \cdot y}{100}$$

Note: Assign positive signs for increases/premiums and negative signs for decreases/discounts.

Worked Example:

• Problem: A retail product's price is increased by 20% and later discounted by 10%. What is the net percentage change from the original price?
• Solution: Let $x = +20$ and $y = -10$.
• $$\text{Net Change} = 20 + (-10) + \frac{20 \times (-10)}{100} = 10 - \frac{200}{100} = 10 - 2 = \mathbf{+8\%} \text{ (Increase)}$$

2.5.1 Concept and Definition

Substitution is an optimization technique where you replace complex abstract algebraic variables with simple, concrete numerical constants to evaluate expressions or verify answer options.

2.5.2 Algebraic Optimization Strategy

Use this method when expressions contain independent constraints, or when verifying the answers directly is faster than traditional algebraic factoring.

Worked Example:

• Problem: If $x^2 - 5x + 6 = 0$, evaluate the value of the algebraic expression $x^3 - 2x$.
• Solution: Find the valid inputs by factoring the given quadratic equation: $(x-2)(x-3) = 0 \implies x = 2 \text{ or } x = 3$.
• Substitute $x=2$ into the target expression:
• $$2^3 - 2(2) = 8 - 4 = \mathbf{4}$$
• (Note: If multiple valid conditions exist, check the available multiple-choice options to identify the matching value).

2.6.1 Arbitrary Operator Mechanics

Defined functions use arbitrary geometric symbols (such as $*$, $\#$, $@$, $\Delta$) to define custom, non-standard mathematical operations. These operations are unique to the problem and must be executed strictly according to their explicit step-by-step formulas.

2.6.2 Multi-Layer Operational Scaling

To evaluate multi-layer custom expressions, apply the standard order of operations (BODMAS) by simplifying inner nested parentheses before working outward.

Worked Example:

• Problem: Let the operation $a \mathbin{\Delta} b$ be defined as $a^2 - b$. Evaluate $(3 \mathbin{\Delta} 2) \mathbin{\Delta} 4$.
• Solution:
• $$\text{First, evaluate the inner parenthesis: } 3 \mathbin{\Delta} 2 = 3^2 - 2 = 9 - 2 = 7$$
• $$\text{Now, substitute this result back: } 7 \mathbin{\Delta} 4 = 7^2 - 4 = 49 - 4 = \mathbf{45}$$

2.7.1 Matrix Elements & Notation

A set is a well-defined collection of distinct mathematical objects.

• Universal Set ($U$): The total population boundary containing all possible elements under analysis.
• Intersection ($A \cap B$): The set containing elements that simultaneously belong to both set $A$ and set $B$.
• Union ($A \cup B$): The set containing all elements that belong to set $A$, set $B$, or both.

2.7.2 Two-Set and Three-Set Formulas

For bivariate (two-set) analytical configurations:

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

$$n(U) = n(A \cup B) + n(A \cup B)'$$

Where $n(A \cup B)'$ represents components outside both target sets.

2.7.3 Structural Venn Diagram Application

               Universal Set (U)
  +---------------------------------------+
  |   Set A               Set B           |
  |   +-------+       +-------+           |
  |   |       |   A   |       |           |
  |   | Only  |   ∩   | Only  |           |
  |   |   A   |   B   |   B   |           |
  |   |       |       |       |           |
  |   +-------+       +-------+           |
  |                                       |
  |        Neither (A U B)'               |
  +---------------------------------------+

Worked Example:

• Problem: In a corporate cohort of 100 managers, 65 are certified in Agile, 45 are certified in PMP, and 20 hold both credentials. How many managers hold neither certification?
• Solution: Apply the standard two-set formula:
• $$n(\text{Agile} \cup \text{PMP}) = 65 + 45 - 20 = 90$$
• $$\text{Neither} = n(U) - n(\text{Agile} \cup \text{PMP}) = 100 - 90 = \mathbf{10}$$

2.8.1 Categorization Models

• Proper Fractions: The numerator is strictly less than the denominator ($\frac{3}{4}$). The absolute value is less than 1.
• Improper Fractions: The numerator is greater than or equal to the denominator ($\frac{5}{3}$).
• Recurring Decimals: Decimals with a digit or sequence of digits that repeats infinitely past the decimal point ($0.333... = 0.\bar{3}$).

2.8.2 Rapid Fraction Comparison Techniques

To sort fractions quickly under exam time constraints, use cross-multiplication rather than converting each value to a long decimal.

$$\text{To compare } \frac{a}{b} \text{ and } \frac{c}{d}, \text{ compute } a \cdot d \text{ and } b \cdot c.$$

$$\text{If } a \cdot d > b \cdot c, \text{ then } \frac{a}{b} > \frac{c}{d}.$$

Worked Example:

• Problem: Identify the larger fraction between $\frac{5}{8}$ and $\frac{7}{11}$.
• Solution: Cross-multiply the terms: $5 \times 11 = 55$ and $8 \times 7 = 56$.
• Since $56 > 55$, it follows that $\frac{7}{11}$ is the larger fraction.

2.9.1 Structural Operations

A matrix is a rectangular array of numbers organized into rows and columns. Matrices can be added or subtracted only if they share identical structural dimensions ($m \times n$).

2.9.2 Determinant Formulations for $2 \times 2$ Formats

The determinant is a scalar value calculated from a square matrix that reveals its structural properties.

$$\text{For Matrix } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \text{ the determinant is } |A| = ad - bc$$

2.9.3 Matrix Inversion & Singularity

A matrix is classified as singular if its determinant is exactly zero ($|A| = 0$). Singular matrices cannot be inverted because the calculation requires dividing by the determinant.

Worked Example:

• Problem: Find the value of $x$ that makes the matrix $M = \begin{pmatrix} x & 4 \\ 3 & 6 \end{pmatrix}$ singular.
• Solution: For $M$ to be singular, its determinant must equal 0.
• $$|M| = (x \cdot 6) - (4 \cdot 3) = 0 \implies 6x - 12 = 0 \implies 6x = 12 \implies x = \mathbf{2}$$

2.10.1 Arithmetic Progression (AP) Structures

An Arithmetic Progression is a sequence of numbers where the difference ($d$) between consecutive terms remains constant.

• General Term ($n^{\text{th}}$ term):
• $$t_n = a + (n-1)d$$
• Sum of $n$ Terms ($S_n$):
• $$S_n = \frac{n}{2} [2a + (n-1)d] = \frac{n}{2} [a + t_n]$$

Where $a$ is the initial term and $d$ is the common difference.

2.10.2 Geometric Progression (GP) Structures

A Geometric Progression is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$).

• General Term ($n^{\text{th}}$ term):
• $$t_n = a \cdot r^{n-1}$$
• Sum of $n$ Terms ($S_n$ for $r < 1$):
• $$S_n = \frac{a(1 - r^n)}{1 - r}$$
• Sum of an Infinite Series ($S_\infty$ valid only if $|r| < 1$):
• $$S_\infty = \frac{a}{1 - r}$$

Worked Example:

• Problem: Compute the sum of the infinite geometric series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$
• Solution: Here, the initial term $a = 1$, and the common ratio $r = \frac{1}{2}$. Since $|r| < 1$, apply the infinite sum formula:
• $$S_\infty = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = \mathbf{2}$$

2.11.1 Direct vs. Inverse Proportionality

The unitary method solves multi-variable workforce problems by first calculating the output capacity of a single unit.

• Direct Variation: More goods cost more money. $\frac{\text{Cost}_1}{\text{Units}_1} = \frac{\text{Cost}_2}{\text{Units}_2}$
• Inverse Variation (Workforce/Time): More workers require less time to complete a fixed task.

2.11.2 Workforce Chain Rule System

To manage complex variations involving workforce size ($M$), operational days ($D$), working hours ($H$), and total work output ($W$), use the unified joint chain rule equation:

$$\frac{M_1 \cdot D_1 \cdot H_1}{W_1} = \frac{M_2 \cdot D_2 \cdot H_2}{W_2}$$

Worked Example:

• Problem: If 12 engineers can configure 4 operational servers in 6 days, how many days will it take 8 engineers to configure 8 identical servers?
• Solution: Identify the variables: $M_1=12, W_1=4, D_1=6$ and $M_2=8, W_2=8, D_2=x$.
• $$\frac{12 \times 6}{4} = \frac{8 \times x}{8} \implies \frac{72}{4} = x \implies x = \mathbf{18 \text{ days}}$$

2.12.1 Foundational Laws of Exponents

Exponents follow strict algebraic laws for operations sharing identical bases:

LawOperational FormulaProduct Rule$a^m \times a^n = a^{m+n}$Quotient Rule$\frac{a^m}{a^n} = a^{m-n}$Power of a Power$(a^m)^n = a^{mn}$Negative Exponent$a^{-m} = \frac{1}{a^m}$Zero Power Rule$a^0 = 1 \quad (\text{for } a \neq 0)$Fractional Index$a^{\frac{1}{n}} = \sqrt[n]{a}$

2.12.2 Exponential Base Equivalence Equations

When solving for an unknown variable in an exponent, simplify the expressions on both sides of the equation until they share an identical base. Once the bases match, you can set the exponents equal to each other.

$$\text{If } a^x = a^y, \text{ then } x = y \quad (\text{where } a \notin \{-1, 0, 1\})$$

Worked Example:

• Problem: Solve for $x$ in the equation: $3^{2x-1} = 27$.
• Solution: Express 27 as a base-3 power: $27 = 3^3$.
• $$3^{2x-1} = 3^3 \implies 2x - 1 = 3 \implies 2x = 4 \implies x = \mathbf{2}$$

2.13.1 Place Value Decimal Framework

For problems involving digit re-orderings, remember that a standard two-digit number cannot be represented as simple multiplication ($xy$). You must write it out using its structural place values.

$$\text{Value of a two-digit number with tens digit } t \text{ and units digit } u = 10t + u$$

$$\text{Value with reversed digits} = 10u + t$$

2.13.2 Timeline Age Shift Calculations

Age word problems require translating word descriptions into linear equations. When shifting across timelines, remember to add or subtract years from every individual involved.

$$\text{If a person's current age is } X, \text{ their age } n \text{ years ago was } X - n, \text{ and their age } n \text{ years from now will be } X + n.$$

⚠️ Critical Invariant Rule: The absolute difference in age between two individuals remains constant forever. It never changes over time.

Worked Example:

• Problem: A father is currently three times as old as his son. Five years ago, the father was four times as old as his son. What are their current ages?
• Solution: Let the son's current age be $S$, so the father's current age is $3S$.
• Formulate the equation representing their ages 5 years ago:
• $$(3S - 5) = 4(S - 5) \implies 3S - 5 = 4S - 20 \implies 20 - 5 = 4S - 3S \implies S = 15$$
• The son is 15 years old, and the father is $3 \times 15 = \mathbf{45 \text{ years old}}$.

2.14.1 Fundamental Principles of Counting

• The Multiplication Principle (AND rule): If an operation can be performed in $m$ ways, followed by a second operation in $n$ ways, both operations can be completed sequentially in $m \times n$ ways.
• The Addition Principle (OR rule): If mutually exclusive events can occur in $m$ or $n$ ways respectively, the choice of either event can occur in $m + n$ ways.

2.14.2 Permutations (Arrangements where order matters)

Permutations calculate the number of unique ways to arrange a subset of items where the order of arrangement is critical.

$$P(n, r) = \frac{n!}{(n-r)!}$$

2.14.3 Combinations (Selections where order does not matter)

Combinations calculate the number of unique ways to select a subset of items where the order of selection does not matter.

$$C(n, r) = \frac{n!}{r!(n-r)!} = \frac{P(n, r)}{r!}$$

Worked Example:

• Problem: How many unique corporate committees of 2 members can be formed from a department of 6 executives?
• Solution: Since a committee is a selection where order does not matter, apply the combination formula:
• $$C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = \mathbf{15 \text{ unique committees}}$$

2.15.1 Simple Interest Equations

Simple interest is calculated solely on the original principal amount.

$$SI = \frac{P \cdot T \cdot R}{100}$$

$$\text{Total Accumulated Amount } (A) = P + SI = P \left(1 + \frac{T \cdot R}{100}\right)$$

2.15.2 Compound Interest Equations

Compound interest calculates interest on both the initial principal and the accumulated interest from prior periods.

$$\text{Total Compounded Amount } (A) = P \left(1 + \frac{R}{100}\right)^T$$

$$\text{Compound Interest } (CI) = A - P = P \left[ \left(1 + \frac{R}{100}\right)^T - 1 \right]$$

2.15.3 Profit & Loss and Cost Basis Formulas

Profit and loss metrics measure financial returns.

⚠️ Profit and loss percentages are always calculated using the Cost Price (CP) as the base, unless a problem explicitly states otherwise.

$$\text{Profit} = \text{Selling Price (SP)} - \text{Cost Price (CP)} \quad (\text{when } SP > CP)$$

$$\text{Loss} = \text{Cost Price (CP)} - \text{Selling Price (SP)} \quad (\text{when } CP > SP)$$

$$\text{Profit Percentage } (\%) = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100$$

$$\text{Selling Price Formulas:} \quad SP = \text{CP} \times \left( \frac{100 + \text{Profit \%}}{100} \right) = \text{CP} \times \left( \frac{100 - \text{Loss \%}}{100} \right)$$

Commission, Taxation, and VAT

2.16.1 Commission and Brokerage Calculation

Commissions are performance incentives calculated as a percentage of total sales volume.

$$\text{Commission Earned} = \text{Total Sales Volume} \times \text{Commission Rate \%}$$

2.16.2 Flat vs. Progressive Income Taxation Mechanics

Taxation structures apply specific rates to income brackets. Progressive taxation divides income into different bands, with higher income bands taxed at higher percentage rates.

2.16.3 Value Added Tax (VAT) & Sequential Discounting

Value Added Tax (VAT) is a consumption tax added to the net value of a product.

⚠️ Critical Calculation Rule: VAT is always calculated on the net selling price after all trade discounts have been subtracted from the Marked Price (MP).

  [ Marked Price (MP) ]
           │
           ▼  (Subtract Discount: MP × Discount %)
  [ Net Selling Price (SP) ]
           │
           ▼  (Add VAT: SP × VAT %)
  [ Final Consumer Price ]

Worked Example:

• Problem: An enterprise software package has a Marked Price of Rs. 10,000. The developer offers a 10% discount, followed by a 13% VAT. What is the final price paid by the customer?
• Solution:
• $$\text{Step 1: Compute SP after discount: } 10,000 \times (1 - 0.10) = 10,000 \times 0.90 = \text{Rs. } 9,000$$
• $$\text{Step 2: Apply 13% VAT to the discounted price: } 9,000 \times (1 + 0.13) = 9,000 \times 1.13 = \mathbf{\text{Rs. } 10,170}$$

2.17.1 Mathematical Foundations

• Highest Common Factor (HCF): The largest positive integer that divides two or more integers without leaving a remainder.
• Lowest Common Multiple (LCM): The smallest positive integer that is a multiple of two or more integers.

2.17.2 The Product Property Rule

For any two positive integers $a$ and $b$, the product of the numbers is equal to the product of their HCF and LCM.

$$a \cdot b = \text{HCF}(a, b) \times \text{LCM}(a, b)$$

2.17.3 Fractions HCF/LCM Calculations

When computing the factors and multiples of fractions, apply these explicit structural equations:

$$\text{HCF of Fractions} = \frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$$

$$\text{LCM of Fractions} = \frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$$

2.18.1 Classical Probability Probability Space Definition

Probability measures the likelihood of a specific event occurring within a defined sample space of equally likely outcomes.

$$P(E) = \frac{\text{Number of Favorable Outcomes } n(E)}{\text{Total Number of Outcomes in Sample Space } n(S)}$$

The value of $P(E)$ is constrained by the boundary definition: $0 \le P(E) \le 1$.

2.18.2 Mutually Exclusive vs. Independent Events

• Mutually Exclusive Events: Events that cannot happen at the same time ($A \cap B = \emptyset$).
• $$P(A \cup B) = P(A) + P(B)$$
• Independent Events: The occurrence of one event does not affect the probability of the other event.
• $$P(A \cap B) = P(A) \cdot P(B)$$

Worked Example:

• Problem: A card is drawn at random from a standard 52-card deck. What is the probability that the card is either a King or a Queen?
• Solution: The events are mutually exclusive (a card cannot be a King and a Queen simultaneously).
• $$P(\text{King}) = \frac{4}{52}, \quad P(\text{Queen}) = \frac{4}{52}$$
• $$P(\text{King} \cup \text{Queen}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \mathbf{\frac{2}{13}}$$

2.19.1 Canonical Standard Form Layout

A quadratic equation is a polynomial equation of the second degree, expressed in standard form as:

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

Where $a, b, c \in \mathbb{R}$ and $a \neq 0$.

2.19.2 The Quadratic Formula & Discriminant Analysis

The two roots ($\alpha, \beta$) of the equation can be calculated using the quadratic formula:

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

The term inside the square root is the discriminant ($D = b^2 - 4ac$). Its value reveals the nature of the roots:

• If $D > 0$: The roots are real and distinct.
• If $D = 0$: The roots are real and identical ($\alpha = \beta$).
• If $D < 0$: The roots are complex/imaginary.

2.19.3 Root Vieta Symmetric Properties

Without solving for the individual roots, you can determine their structural relationships directly from the coefficients: