Comprehensive Syllabus for Numerical Ability

Number System - This topic serves as the baseline for numerical ability, involving the classification and properties of numbers. - Types of numbers include natural numbers, whole numbers, integers ( $Z$ ), rational numbers ( $Q$ ), irrational numbers, and real numbers ( $R$ ). - Sub-topics often include divisibility rules, unit digit calculations, remainder theorems, and the identification of prime and composite numbers.
L.C.M & H.C.F - Least Common Multiple (L.C.M.): The smallest positive integer that is divisible by two or more numbers. - Highest Common Factor (H.C.F.): The largest positive integer that divides two or more numbers without leaving a remainder. - Key Formula: For any two numbers $a$ and $b$ , the relationship is defined as $HCF(a, b) \times LCM(a, b) = a \times b$ .

Percentages - Operations involving fractions with a denominator of 100. It is used to express how one quantity relates to another in terms of hundreds. - Formula: $\text{Percentage} = \frac{\text{Value}}{\text{Total Value}} \times 100$ .
Profit and Loss - Calculations based on Cost Price ( $CP$ ) and Selling Price ( $SP$ ) of an item. - Formula for Profit: $SP - CP$ . - Formula for Loss: $CP - SP$ . - Profit Percentage: $\frac{Profit}{CP} \times 100$ .
Simple Interest & Compound Interest - Simple Interest (S.I.): Interest calculated only on the principal amount ( $P$ ) over time ( $T$ ) at a specific rate ( $R$ ). - Formula: $SI = \frac{P \times R \times T}{100}$ . - Compound Interest (C.I.): Interest calculated on the principal and also on the accumulated interest of previous periods. - Formula for Amount ( $A$ ): $A = P(1 + \frac{R}{100})^T$ . - Interest calculation: $CI = A - P$ .
Averages - The sum of all observations divided by the total number of observations. - Formula: $\text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}}$ .

Ratio & Proportion - Ratio: A comparison of two quantities by division, expressed as $a:b$ or $\frac{a}{b}$ . - Proportion: An equation that states two ratios are equal, expressed as $a:b = c:d$ .
Partnership - Calculations involving the distribution of profits among business partners based on their initial investment amounts and the duration for which the capital was invested. - Profit Sharing Ratio: $P_1 : P_2 = (I_1 \times T_1) : (I_2 \times T_2)$ , where $I$ represents Investment and $T$ represents Time.
Problems on Ages - Algebraic problems designed to find the current, past, or future ages of individuals based on given linear equations or ratios.
Allegation & Mixtures - A rule that enables the calculation of the price or ratio of a mixture given the prices/quantities of individual components. - Used when mixing two items of different values to reach a mean price.

Time and Distance - Fundamental dynamics of motion relating speed ( $S$ ), distance ( $D$ ), and time ( $T$ ). - Basic formula: $S = \frac{D}{T}$ . - Standard conversions: $1 \text{ km/h} = \frac{5}{18} \text{ m/s}$ and $1 \text{ m/s} = \frac{18}{5} \text{ km/h}$ .
Problems on Trains - Specialized applications of time and distance formulas, often accounting for the length of the train and the length of stationary objects like platforms or moving objects like other trains. - Relative Speed: If two trains move in the same direction, speed is $\text{Speed}_1 - \text{Speed}_2$ . If moving in opposite directions, speed is $\text{Speed}_1 + \text{Speed}_2$ .
Boats and Streams - Problems involving the speed of a boat in still water versus moving water (current). - Downstream Speed: Speed of boat + Speed of stream. - Upstream Speed: Speed of boat - Speed of stream.

Time and Work - Calculating the time required to complete a task based on the efficiency of workers. - Concept of Work Rate: If a person completes work in $n$ days, their one-day work is $\frac{1}{n}$ . - Chain Rule Formula: $\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$ , where $M$ is men, $D$ is days, $H$ is hours, and $W$ is work.
Pipes & Cisterns - Similar to time and work, but applied to filling (inlet pipes) or emptying (outlet/leakage pipes) a tank or cistern.

Mensuration - The study of geometric figures and their parameters. - 2D Mensuration: Calculation of Area and Perimeter for figures like squares, rectangles, triangles, and circles. - 3D Mensuration: Calculation of Volume, Lateral Surface Area, and Total Surface Area for solids like cubes, cuboids, cylinders, cones, and spheres.
Probability - The measure of the likelihood that an event will occur. - Formula: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ . - Includes concepts from permutations and combinations to determine the total sample space and event occurrences.