Mathematics Number System and Percentages

Types of Numbers: Natural numbers, Whole numbers, Integers, Rational numbers, Irrational numbers, Real numbers.
Divisibility Rules:
- A number is divisible by 2 if it is even.
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 5 if it ends in 0 or 5.
LCM and GCD:
- LCM (Least Common Multiple) is the smallest multiple common to two or more numbers.
- GCD (Greatest Common Divisor) is the largest factor common to two or more numbers.
Fractions: Simplification, addition, subtraction, multiplication, and division of fractions.

Definition: A percentage is a fraction of 100.
Conversion:
- Percentage to decimal: Divide by 100 (e.g., 25% = 0.25).
- Decimal to percentage: Multiply by 100 (e.g., 0.75 = 75%).
Calculations:
- To find a percentage of a number: (Percentage/100) × Number.
- Increase or decrease a number by a percentage involves finding the percentage and adding/subtracting it.
Successive Percentage Change: Use the formula (1 + p1/100) × (1 + p2/100) for calculating successive improvements.

Basic Concepts:
- Cost Price (CP): Price at which an item is bought.
- Selling Price (SP): Price at which an item is sold.
- Profit = SP - CP; Loss = CP - SP.
Profit Percentage: (Profit/CP) × 100.
Loss Percentage: (Loss/CP) × 100.
Discount: Reduction from the marked price; Discount = Marked Price - Selling Price.

Work Efficiency:
- Work done by one person in one day = 1/(Number of days taken).
- If A can do a job in x days and B in y days, A and B together can do the job in (xy)/(x+y) days.
Work Problems:
- Use the formula: Total Work = Number of Workers × Time Taken.
- Incorporate conditions like days off, work done in different rates, etc.

Types of Data Representation:
- Bar graphs, Pie charts, Line graphs, Tables.
Key Skills:
- Ability to extract information from visual data.
- Calculation of averages, percentages, and ratios from data sets.
Analyzing Trends: Identifying increases, decreases, and patterns in data.
Common Questions:
- Finding totals, averages, and making comparisons between different data sets.

Types of Numbers: Includes natural numbers (1, 2, 3...), whole numbers (0, 1, 2...), integers (..., -1, 0, 1...), rational numbers (numbers that can be expressed as fractions), irrational numbers (non-repeating, non-terminating decimals), and real numbers (all rational and irrational numbers).
Divisibility Rules:
- Divisibility by 2: A number is even.
- Divisibility by 3: The sum of the digits is divisible by 3.
- Divisibility by 5: The number ends in 0 or 5.
LCM and GCD:
- LCM (Least Common Multiple) is the smallest non-zero multiple that is common to two or more numbers.
- GCD (Greatest Common Divisor) is the largest integer that divides two or more numbers without a remainder.
Fractions: Operations include simplification, and calculating addition, subtraction, multiplication, and division of fractions.

Definition: Represents a value as a fraction of 100, useful for comparing relative sizes.
Conversion:
- Convert percentage to decimal by dividing by 100 (e.g., 25% becomes 0.25).
- Convert decimal to percentage by multiplying by 100 (e.g., 0.75 becomes 75%).
Calculations:
- To find a percentage of a number, multiply the percentage by the number and divide by 100.
- Increasing a number by a percentage involves calculating the percentage and adding it to the original number; decreasing involves subtracting.
Successive Percentage Change: For multiple changes, use the formula (1 + p1/100) × (1 + p2/100) to calculate cumulative percentage changes.

Basic Concepts:
- Cost Price (CP) is the purchase price of an item, while Selling Price (SP) is the price at which the item is sold.
- Profit is calculated as SP minus CP, while Loss is calculated as CP minus SP.
Profit Percentage: Calculated as (Profit/CP) × 100 to determine profit relative to the cost price.
Loss Percentage: Calculated as (Loss/CP) × 100 to determine loss relative to the cost price.
Discount: Discount is the difference between the marked price and the selling price.

Work Efficiency:
- The daily work output of one person can be calculated as the reciprocal of the total days they take to complete a job.
- For two workers A and B, if A completes a job in x days and B in y days, their combined efficiency allows them to complete the job in (xy)/(x+y) days.
Work Problems:
- Total Work can be expressed as the product of the total number of workers and the time taken, allowing adjustment for different rates of work and days off.

Types of Data Representation: Common forms include bar graphs (useful for comparison), pie charts (showing parts of a whole), line graphs (displaying trends over time), and tables (organizing data for clarity).
Key Skills:
- Ability to extract and interpret information from visual data representations.
- Skills in calculating averages, percentages, and ratios based on data sets.
Analyzing Trends: Recognizing patterns such as increases and decreases in data allows for informed conclusions.
Common Questions: Typical tasks include finding totals, averages, and making comparisons across various data sets.