Number - Percentages, profit and loss, simple and compound interest

Percentages represent parts of a whole as a fraction of $100$. Visually, this can be understood using a 'hundred square'—a $10 \times 10$ grid where each small square equals $1\%$. To convert any fraction to a percentage, you multiply the fraction by $100$. For example, $\frac{3}{4}$ of a shape shaded is equivalent to $75\%$.

Percentage change describes how much a value has increased or decreased relative to its original amount. This is often visualized using a bar model where the original value is a bar representing $100\%$. An increase adds a smaller bar to the end, while a decrease shades out or removes a portion of the original bar. The new value is found by multiplying the original by a multiplier like $(1 + \frac{\text{percentage}}{100})$ for growth.

Profit and Loss are financial measures based on the Cost Price ($CP$) and Selling Price ($SP$). On a horizontal number line where the $CP$ is the starting point, a movement to the right (where $SP > CP$) represents a profit, while a movement to the left (where $SP < CP$) represents a loss. Profit and loss are usually expressed as a percentage of the original Cost Price.

Simple Interest is interest calculated only on the initial principal amount ($P$) for the entire duration. This results in the interest amount being the same every year. When plotted on a coordinate plane with 'Time' on the $x$-axis and 'Total Amount' on the $y$-axis, simple interest creates a straight, diagonal line (linear growth) starting from the principal value.

Compound Interest is interest calculated on the initial principal and also on the accumulated interest of previous periods. Visually, this creates a 'snowball effect' represented by an exponential curve on a graph. Unlike the straight line of simple interest, the compound interest curve gets steeper over time as the 'interest on interest' adds up.

Reverse Percentages involve finding the original value ($100\%$) after a percentage change has been applied. This can be visualized as a flowchart: Original Value $\rightarrow$ [Multiply by Change Factor] $\rightarrow$ New Value. To find the original, you work backward: New Value $\rightarrow$ [Divide by Change Factor] $\rightarrow$ Original Value. It is a common mistake to simply apply the percentage to the new value; you must always divide by the multiplier.