Key Points

To convert a ratio such as $a:b$ into a percentage, first express it as a fraction $\frac{a}{b}$ and then multiply by 100. For example, the ratio $3:4$ is equivalent to $\frac{3}{4} \times 100 = 75\%$.

Percentage change is calculated based on the original value. The formula is $\text{Percentage Change} = \frac{\text{Change in Value}}{\text{Original Value}} \times 100$. The new value is found by adding (for increase) or subtracting (for decrease) the change from the original value.

Discount is a reduction given on the Marked Price (MP). The formula to calculate the discount amount is $\text{Discount} = \text{Marked Price} - \text{Sale Price}$.

The discount percentage is always calculated with respect to the Marked Price. The formula is $\text{Discount} \% = \frac{\text{Discount}}{\text{Marked Price}} \times 100$.

The Sale Price (SP) is the price a customer pays after the discount. It can be calculated as $\text{SP} = \text{Marked Price} - \text{Discount}$. Alternatively, for a discount of $D\%$, $\text{SP} = \text{MP} \times \left(1 - \frac{D}{100}\right)$.

Taxes like Sales Tax, VAT, or GST are charged on the selling price and added to the bill. The final bill amount is calculated as $\text{Bill Amount} = \text{Cost} + \text{Tax}$. The tax amount is $\text{Tax} = \frac{\text{Tax} \%}{100} \times \text{Cost}$.

To find the original price before tax was added, use the formula: $\text{Original Price} = \frac{\text{Final Price Including Tax}}{1 + \frac{\text{Tax Rate}}{100}}$. For a 10% tax, you divide the final price by 1.10.

Simple Interest (SI) is calculated only on the original principal amount. The formula is $\text{SI} = \frac{P \times R \times T}{100}$, where $P$ is the Principal, $R$ is the annual rate of interest, and $T$ is the time in years.

Compound Interest (CI) is calculated on the principal and the accumulated interest from previous periods. This is often called 'interest on interest' and causes the amount to grow faster than with simple interest.

When interest is compounded annually, the total amount ($A$) after $n$ years is given by the formula $A = P\left(1 + \frac{R}{100}\right)^n$. Here, $P$ is the principal, $R$ is the annual interest rate, and $n$ is the number of years.

The Compound Interest (CI) is the total interest earned over the period. It is calculated by subtracting the original principal from the final amount: $\text{CI} = \text{Amount} - \text{Principal}$ or $\text{CI} = A - P$.

The compound interest formula is used to model population growth. The future population is calculated as $\text{Future Population} = \text{Current Population} \times \left(1 + \frac{\text{Growth Rate}}{100}\right)^n$.

Depreciation is the reduction in the value of an asset over time. The depreciated value ($V$) is calculated using a modified formula: $V = P\left(1 - \frac{R}{100}\right)^n$, where $P$ is the initial value and $R$ is the rate of depreciation.