Percentage and Percentage Change

• Percentage and percentage change are fundamental concepts in mathematics and are widely used in various real-life applications, especially in business and finance.
• The percentage represents a fraction or ratio with a denominator of 100.
• It is denoted by the symbol "%".

Basic Percentage Calculation

• To express a number as a percentage, multiply it by 100.
• Percentage = (Value / Total Value) * 100
• For example, if 20 out of 50 students passed an exam, the percentage of students who passed is (20/50) * 100 = 40%.

Percentage Change

• Percentage change calculates the extent to which a quantity changes over time.
• The formula for percentage change is: [(New Value - Old Value) / Old Value] * 100
• A positive percentage change indicates an increase, while a negative percentage change indicates a decrease.

Cost Price (CP)

• The cost price is the price at which an item is purchased.
• It includes the manufacturer's price plus any additional expenses such as transportation, taxes, and other overheads.

Selling Price (SP)

• The selling price which an item is sold
• This is the price the customer pays

Profit

• Profit occurs when the selling price (SP) is greater than the cost price (CP).
• Profit = Selling Price (SP) - Cost Price (CP)

Loss

• Loss occurs when the cost price (CP) is greater than the selling price (SP).
• Loss = Cost Price (CP) - Selling Price (SP)

Profit Percentage

• Profit percentage expresses the profit as a percentage of the cost price.
• Profit Percentage = (Profit / Cost Price) * 100
• For example, if an item is bought for $50 (CP) and sold for $75 (SP), the profit is $25, and the profit percentage is (25/50) * 100 = 50%.

Loss Percentage

• Loss percentage expresses the loss as a percentage of the cost price.
• Loss Percentage = (Loss / Cost Price) * 100
• For example, if an item is bought for $50 (CP) and sold for $40 (SP), the loss is $10, and the loss percentage is (10/50) * 100 = 20%.

Markup

• Markup is the amount added to the cost price to determine the selling price.
• Markup = Selling Price - Cost Price
• It is often expressed as a percentage of the cost price.
• Markup Percentage = (Markup / Cost Price) * 100

Discount

• Discount is a reduction in the usual selling price of an item.
• Discount = Marked Price (Original Price) - Selling Price
• It is often expressed as a percentage of the marked price.
• Discount Percentage = (Discount / Marked Price) * 100

Relationships

• If the profit percentage is given, the selling price can be calculated as: SP = CP * [1 + (Profit Percentage / 100)]
• If the loss percentage is given, the selling price can be calculated as: SP = CP * [1 - (Loss Percentage / 100)]
• If a discount percentage is given off the marked price (MP), the selling price is: SP = MP * [1 - (Discount Percentage / 100)]

Successive Discounts

• When two or more discounts are applied one after the other; first discount is applied to the marked price, and the second discount is applied to the reduced price after the first discount.
• For successive discounts of x% and y%, the effective discount percentage is: [x + y - (xy/100)]%

Example Calculations

• Original Price: $200, Increase by 10%: $200 * 0.10 = $20, New Price = $200 + $20 = $220
• Original Price: $200, Decrease by 10%: $200 * 0.10 = $20, New Price = $200 - $20 = $180
• Cost Price: $100, Selling Price: $120, Profit: $120 - $100 = $20, Profit Percentage: ($20 / $100) * 100 = 20%
• Cost Price: $100, Selling Price: $80, Loss: $100 - $80 = $20, Loss Percentage: ($20 / $100) * 100 = 20%
• Marked Price: $150, Discount: 15%, Discount Amount: $150 * 0.15 = $22.50, Selling Price: $150 - $22.50 = $127.50

Applications

• Percentage and percentage change calculations are essential in determining profitability, understanding financial statements, and evaluating investment options.
• Retailers use them to set prices, offer discounts, and analyze sales data.
• Financial institutions use them to calculate interest rates, loan payments, and investment returns.
• They are used to determine growth rates in economics, such as GDP growth, inflation rates, and unemployment rates.
• Used to express changes in stock prices.

Percentage and percentage change are fundamental mathematical concepts used in real-life applications, especially in business and finance. Percentage represents a fraction with a denominator of 100, denoted by '%'. The percentage change calculates the extent to which a quantity changes over time.