Percentages serve as a fundamental tool in expressing fractions and understanding changes in quantities across various contexts. This section explores the practical applications of percentages in more complex scenarios such as financial growth, discounts, and interest calculations.

Percentage Increase and Decrease

The percentage change in a quantity can indicate an increase or decrease, essential for assessing growth, decline, or adjustments in values over time.

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Percentage Increase: The formula to calculate the increase in a value as a percentage of its original amount is:

Example: If a product's price rises from £200 to £250, the percentage increase is calculated as:

Percentage Decrease: Similarly, to calculate the decrease, the formula is:

Example: If another product's price decreases from £150 to £120, the percentage decrease is:

Simple Interest

Simple interest is interest earned or paid on the original principal only. Simple interest is calculated on the principal amount of a loan or investment, remaining constant over the period.

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Formula:

Where $I$ is the interest, $P$ is the principal amount, $R$ is the annual rate of interest, and $T$ is the time in years.

Example: Calculate the simple interest on £1,000 at an annual rate of 5% over 3 years.

Solution:

Compound Interest

Compound interest, in contrast, is calculated on the principal and any accumulated interest, leading to exponential growth.

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Formula:

Where $A$ is the future value of the investment/loan, including interest, $P$ is the principal amount, $r$ is the annual interest rate (decimal), $n$ is the number of times the interest is compounded per year, and $t$ is the time the money is invested or borrowed for, in years.

Example: Calculate the compound interest on £1,000 at an annual interest rate of 5% compounded annually over 3 years.

Solution:

The interest earned would be £1157.63 - £1000 = £157.63.

Real-life Applications

Understanding percentages enables informed decision-making in various scenarios, such as:

Profit and Loss

Determining the profitability of transactions in business.

Example: A retailer buys a batch of electronics for £5,000 and sells it for £6,500.

Solution:

The percentage profit is:

Discounts

Calculating the actual price after applying a discount.

Example: A £200 jacket is on sale with a 25% discount.

Solution:

The sale price is:

Banking and Investments

Understanding simple and compound interest helps in evaluating the returns on savings accounts and investments, crucial for personal finance management.