Key Points

A percentage is a fraction with a denominator of 100. The symbol '%' means 'per cent' or 'out of one hundred'. For example, $25\%$ means 25 out of 100.

To convert a fraction to a percentage, multiply the fraction by 100 and add the percent symbol. For a fraction $\frac{a}{b}$, the percentage is $(\frac{a}{b} \times 100)\%$.

To convert a percentage to a fraction, write the number over 100 and simplify if possible. The formula is $x\% = \frac{x}{100}$.

Fractions, decimals, and percentages are interchangeable. To convert a decimal to a percentage, multiply by 100. For example, the fraction $\frac{1}{2}$ is equal to the decimal $0.5$ and the percentage $50\%$.

To find a percentage of a given quantity, convert the percentage to a fraction or decimal and then multiply. The formula is $y\%$ of a value $z$ is calculated as $\frac{y}{100} \times z$.

Percentage increase shows how much a quantity has grown relative to its original value. The formula is $\text{Percentage Increase} = \frac{\text{Amount of Increase}}{\text{Original Amount}} \times 100$.

Percentage decrease shows how much a quantity has reduced relative to its original value. The formula is $\text{Percentage Decrease} = \frac{\text{Amount of Decrease}}{\text{Original Amount}} \times 100$.

Profit and loss percentages are always calculated based on the Cost Price (CP). The formulas are $\text{Profit} \% = \frac{\text{Profit}}{CP} \times 100$ and $\text{Loss} \% = \frac{\text{Loss}}{CP} \times 100$.

To find the Selling Price (SP), use the Cost Price (CP) and the profit or loss percentage. For a profit of $p\%$, $SP = CP \times (1 + \frac{p}{100})$. For a loss of $l\%$, $SP = CP \times (1 - \frac{l}{100})$.

A discount is a reduction given on the Marked Price (MP). The Selling Price (SP) is the price after the discount, calculated as $SP = MP - \text{Discount}$. The discount percentage is always calculated on the MP.

Taxes like GST are added to the price of an item. The final price is found by adding the tax amount. $\text{Final Price} = \text{Original Price} \times (1 + \frac{\text{Tax}\%}{100})$.

Simple interest is calculated only on the original principal amount. The total Amount $A$ after $t$ years is $A = P(1 + rt)$, where $P$ is the principal, $r$ is the annual rate as a decimal, and $t$ is time in years.

In compound interest, interest is added to the principal for each period, and future interest is calculated on this new principal. The total Amount $A$ after $t$ periods is $A = P(1 + r)^t$, where $r$ is the rate per period.

Depreciation is the decrease in value of an asset over time. The final value $V$ after $t$ years is $V = P(1 - r)^t$, where $P$ is the initial price and $r$ is the annual depreciation rate as a decimal.

Successive percentage changes, like a $30\% + 20\%$ discount, are applied one after another on the new value. This is not the same as a single change of their sum ($50\%$).

Calculating a percentage of a number is commutative, meaning $x\%$ of $y$ is always equal to $y\%$ of $x$. For example, $25\%$ of $40$ ($10$) is the same as $40\%$ of $25$ ($10$).

If a percentage of an amount is known, the original amount can be found. If $p\%$ of a number is $x$, the original number is $x \div \frac{p}{100}$ or $x \times \frac{100}{p}$.