Quantitative Aptitude: Number Systems

• Quantitative aptitude is a core skill assessed in various competitive exams
• It measures a candidate's ability to understand and apply mathematical concepts

Number Systems

• Number systems involve different types of numbers and their properties

Types of Numbers

• Natural numbers are positive integers starting from 1 (1, 2, 3, ...)
• Whole numbers include all natural numbers and 0 (0, 1, 2, 3, ...)
• Integers include all whole numbers and their negative counterparts (... -3, -2, -1, 0, 1, 2, 3, ...)
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0 (e.g., 1/2, -3/4, 5)
• Irrational numbers cannot be expressed as a simple fraction (e.g., √2, π)
• Real numbers include both rational and irrational numbers
• Prime numbers are numbers greater than 1 that have only two factors: 1 and the number itself (e.g., 2, 3, 5, 7, 11)
• Composite numbers are numbers greater than 1 that have more than two factors (e.g., 4, 6, 8, 9)

Divisibility Rules

• Divisibility rules provide quick ways to determine if a number is divisible by another number without performing division
• A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8)
• A number is divisible by 3 if the sum of its digits is divisible by 3
• A number is divisible by 4 if the number formed by its last two digits is divisible by 4
• A number is divisible by 5 if its last digit is 0 or 5
• A number is divisible by 6 if it is divisible by both 2 and 3
• A number is divisible by 8 if the number formed by its last three digits is divisible by 8
• A number is divisible by 9 if the sum of its digits is divisible by 9
• A number is divisible by 10 if its last digit is 0
• A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or divisible by 11

Averages

• Averages provide a measure of central tendency for a set of numbers
• To calculate the average (arithmetic mean), sum all the numbers in the set and divide by the count of numbers

Formulas

• Average = (Sum of observations) / (Number of observations)
• Sum of observations = Average * Number of observations
• Weighted Average is used when different data points have different weights
• Weighted Average = (Σ(Weight * Value)) / (Σ Weight)

Percentages

• Percentages express a number as a fraction of 100
• Percentage = (Value / Total Value) * 100

Percentage Change

• Percentage Increase = [(New Value - Original Value) / Original Value] * 100
• Percentage Decrease = [(Original Value - New Value) / Original Value] * 100

Applications of Percentages

• Profit and Loss calculation involves determining the percentage gain or loss on a transaction
• Discount calculation finds the reduced price after applying a percentage discount
• Simple Interest calculation uses a percentage rate to determine the interest earned over time
• Compound Interest calculation involves earning interest on the principal and accumulated interest

Ratio and Proportion

• Ratios compare the relative sizes of two or more quantities
• A ratio is expressed as a:b, where a and b are quantities of the same unit
• A proportion states that two ratios are equal, a:b = c:d

Types of Proportions

• Direct Proportion: If two quantities increase or decrease together, they are directly proportional
• Inverse Proportion: If one quantity increases while the other decreases, they are inversely proportional
• Compound Proportion: Problems involving multiple quantities and their relationships

Profit and Loss

• Profit and loss are important concepts in business and trade
• Cost Price (CP) is the price at which an item is purchased
• Selling Price (SP) is the price at which an item is sold

Formulas

• Profit = SP - CP (if SP > CP)
• Loss = CP - SP (if CP > SP)
• Profit Percentage = (Profit / CP) * 100
• Loss Percentage = (Loss / CP) * 100

Discount

• Discount is a reduction in the marked price of an item
• Discount = Marked Price - Selling Price
• Discount Percentage = (Discount / Marked Price) * 100

Simple and Compound Interest

• Interest is the amount earned on a principal sum of money
• Simple Interest (SI) is calculated only on the principal amount
• Compound Interest (CI) is calculated on the principal and accumulated interest

Simple Interest Formulas

• SI = (P * R * T) / 100
• Where P = Principal, R = Rate of Interest, T = Time in years

Compound Interest Formulas

• A = P(1 + R/100)^T
• CI = A - P
• Where A = Amount after T years, P = Principal, R = Rate of Interest, T = Time in years

Time and Work

• Time and work problems involve calculating the time taken to complete a task by one or more individuals
• If a person can do a piece of work in n days, then their 1 day's work is 1/n

Formulas

• If A can do a work in x days and B can do the same work in y days, then together they can do the work in (xy) / (x + y) days
• Work Done = Efficiency * Time

Time and Distance

• Time and distance problems involve calculating the relationship between speed, time, and distance
• Speed = Distance / Time
• Distance = Speed * Time
• Time = Distance / Speed

Conversions

• km/hr to m/s: multiply by 5/18
• m/s to km/hr: multiply by 18/5

Relative Speed

• When two objects move in the same direction, their relative speed is the difference of their speeds
• When two objects move in opposite directions, their relative speed is the sum of their speeds

Permutations and Combinations

• Permutations and combinations are used to count the number of possible arrangements or selections of a set of items

Permutations

• A permutation is an arrangement of items in a specific order
• nPr = n! / (n - r)!
• Where n is the total number of items, and r is the number of items to arrange

Combinations

• A combination is a selection of items without regard to order
• nCr = n! / [r! * (n - r)!]
• Where n is the total number of items, and r is the number of items to select

Data Interpretation

• Data Interpretation involves understanding and analyzing data presented in various formats, such as tables, charts, and graphs

Types of Charts

• Bar Charts display data using rectangular bars
• Pie Charts display data as proportions of a circle
• Line Charts display data points connected by lines
• Tables arrange data in rows and columns

Analysis

• Read the titles and labels carefully to understand what the data represents
• Look for trends, patterns, and comparisons in the data
• Calculate percentages, ratios, and averages to draw meaningful conclusions