Quantitative Aptitude: Basic Math Concepts

• Quantitative aptitude is the ability to understand and work with numbers
• It involves skills such as mathematical problem-solving, data interpretation, and logical reasoning
• It is often assessed in standardized tests for employment or academic admissions

Basic Mathematical Concepts

• Number systems: Understanding different types of numbers (integers, fractions, decimals, etc.) is fundamental
• Integers include positive and negative whole numbers and zero
• Fractions represent parts of a whole and consist of a numerator and a denominator
• Decimals are another way to represent fractions, using a decimal point
• Percentages are fractions expressed as a part of 100
• Arithmetic operations: Addition, subtraction, multiplication, and division are the basic operations
• These operations apply to all types of numbers
• Order of operations: Follow the PEMDAS/BODMAS rule
• This rule clarifies the sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction
• Simplification: Simplify expressions using algebraic identities and rules to reduce complexity
• Algebraic identities like (a + b)^2 = a^2 + 2ab + b^2 can greatly simplify calculations

Percentage

• Definition: A percentage is a way of expressing a number as a fraction of 100
• Calculation: To find a percentage of a number, multiply the number by the percentage (expressed as a decimal)
• Percentage increase/decrease: ((New Value - Old Value) / Old Value) * 100
• Percentage change: Illustrates the extent to which a quantity gains or loses value
• Successive percentage change: If a value is changed by x% and then by y%, the effective change is (x + y + (xy/100))%
• Useful for problems involving compound interest or discounts

Ratio and Proportion

• Ratio: A ratio compares two quantities
• It can be expressed as a:b or a/b
• Proportion: A proportion states that two ratios are equal (a/b = c/d)
• Direct proportion: If a increases, b increases (a ∝ b)
• Inverse proportion: If a increases, b decreases (a ∝ 1/b)
• Compound proportion: Problems involving multiple ratios and proportions solved using combined relationships

Average

• Definition: Average is the sum of all values divided by the number of values
• Calculation: Average = (Sum of values) / (Number of values)
• Weighted average: Assigns different weights to different values
• Weighted average = (Σ(Value * Weight)) / (Σ Weights)
• Average speed: Total distance traveled divided by the total time taken

Time and Work

• Basic concept: Relates to the amount of work done by individuals or groups in a certain time
• Formula: Work = Rate * Time
• If a person can do a piece of work in 'n' days, their rate of work is 1/n
• Combined work: If A can do a work in x days and B in y days, together they can do the work in (xy)/(x+y) days
• Work and wages: Problems involving division of wages based on work done individually

Time and Distance

• Basic formula: Distance = Speed * Time
• Units: Ensure consistency in units (km/h, m/s)
• Conversion: 1 km/h = 5/18 m/s and 1 m/s = 18/5 km/h
• Average speed: Total distance divided by total time
• Relative speed:
  • When two objects move in the same direction: Relative speed = |Speed of A - Speed of B|
  • When two objects move in opposite directions: Relative speed = Speed of A + Speed of B
• Problems on trains, boats, and streams: Application of relative speed concepts

Simple and Compound Interest

• Simple interest (SI): Calculated only on the principal amount
• Formula: SI = (P * R * T) / 100, where P = Principal, R = Rate of interest, T = Time
• Compound interest (CI): Calculated on the principal and accumulated interest
• Formula: A = P(1 + R/100)^T, where A = Amount, P = Principal, R = Rate of interest, T = Time
• CI annually, semi-annually, quarterly: Adjust the rate and time accordingly
• Difference between CI and SI: Problems based on finding the difference for a given period

Profit and Loss

• Cost Price (CP): The price at which an item is purchased
• Selling Price (SP): The price at which an item is sold
• Profit: SP > CP, Profit = SP - CP
• Loss: SP < CP, Loss = CP - SP
• Profit percentage: (Profit / CP) * 100
• Loss percentage: (Loss / CP) * 100
• Discount: Reduction on the marked price
• Marked Price (MP): The price at which an item is listed
• Discount = MP - SP
• Discount percentage: (Discount / MP) * 100

Data Interpretation

• Tables: Extract and interpret data presented in tabular form
• Bar graphs: Compare quantities using bars of different lengths
• Pie charts: Represent data as sectors of a circle, showing proportions
• Line graphs: Show trends and changes in data over time
• Focus on extracting relevant information and making comparisons

Permutation and Combination

• Permutation: Arrangement of objects in a specific order
• Formula: nPr = n! / (n-r)!, where n is the total number of objects and r is the number of objects to be arranged
• Combination: Selection of objects without regard to order
• Formula: nCr = n! / (r! * (n-r)!), where n is the total number of objects and r is the number of objects to be selected
• Application: Problems involving arrangements, selections, and probability

Probability

• Definition: The measure of the likelihood that an event will occur
• Formula: Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
• Basic concepts: Sample space, events, mutually exclusive events
• Types of problems: Coin tosses, dice rolls, card games, and general selection problems

Clocks and Calendars

• Clocks: Problems involving angles between hands, time gained or lost
• Angle between hands: |(11/2)M - 30H|, where M = minutes and H = hours
• Calendars: Problems based on days of the week, leap years, and specific dates

Geometry and Mensuration

• Area and perimeter: Formulas for squares, rectangles, triangles, circles
• Volume and surface area: Formulas for cubes, cuboids, spheres, cylinders, cones
• Pythagorean theorem: a^2 + b^2 = c^2 in a right-angled triangle
• Geometric properties: Properties of triangles, quadrilaterals, and circles