Number - Ratios and Proportions

Understanding Ratios: A ratio is a comparison between two or more quantities of the same kind, usually expressed in the form $a:b$. Visually, a ratio like $2:3$ can be represented by a bar model with $2$ blocks of one color and $3$ blocks of another, showing the relationship between the parts.

Simplifying Ratios: Ratios are simplified by dividing all terms by their Highest Common Factor (HCF) until no further division is possible. For example, the ratio $12:18$ simplifies to $2:3$ after dividing both sides by $6$. Visually, this is like zooming out on a pattern of tiles while the relative proportions of colors remain the same.

Sharing in a Given Ratio: To divide a total quantity into a ratio $a:b$, you must first calculate the total number of parts by adding the terms of the ratio ($a + b$). Then, divide the total quantity by these parts to find the value of 'one part' before multiplying back by $a$ and $b$ respectively.

Direct Proportion: Two quantities are in direct proportion if an increase in one leads to a proportional increase in the other. Visually, a graph of two quantities in direct proportion is always a straight line that passes through the origin $(0,0)$. The relationship is defined by $y = kx$, where $k$ is the constant of proportionality.

Inverse Proportion: Two quantities are in inverse proportion if an increase in one leads to a proportional decrease in the other, such that their product remains constant ($xy = k$). Visually, this relationship is represented by a downward-sloping curve called a hyperbola that approaches the axes but never touches them.

Unit Rate and Unitary Method: The unit rate is a comparison where the second term is $1$, such as $5 \text{ dollars per kg}$. This is useful for 'best buy' problems. Visually, the unit rate is the gradient (steepness) of the line on a direct proportion graph.

Scale Drawings and Map Ratios: Scale is the ratio of the length of a drawing or map to the actual length. A scale of $1:n$ means $1 \text{ unit}$ on the drawing represents $n$ units in real life. Visually, a $1 \text{ cm}$ line on a $1:50,000$ map represents $50,000 \text{ cm}$ (or $500 \text{ m}$) of actual terrain.