Number - Ratio and Proportion

Understanding Ratios: A ratio is a mathematical comparison of two or more quantities of the same kind. For example, if a recipe uses $3$ cups of flour and $2$ cups of sugar, the ratio is $3:2$. Visually, this can be represented using a tape diagram or bar model where $3$ equal-sized blocks represent flour and $2$ identical blocks represent sugar, totaling $5$ parts.

Simplifying Ratios: Ratios are expressed in their simplest form by dividing all terms by their Highest Common Factor (HCF). For instance, the ratio $10:15$ simplifies to $2:3$ by dividing both sides by $5$. Visually, this is like taking a large group of items and grouping them into the smallest possible repeating sets.

Equivalent Ratios: Two ratios are equivalent if they express the same relationship. You can create equivalent ratios by multiplying or dividing both parts of the ratio by the same non-zero number. On a coordinate plane, equivalent ratios $(x, y)$ will always lie on a straight line that passes through the origin $(0,0)$.

Sharing in a Given Ratio: To divide a total quantity into a ratio $a:b$, first calculate the total number of parts by adding $a + b$. Then, find the value of one 'part' by dividing the total quantity by the sum of parts. Finally, multiply the value of one part by $a$ and $b$ respectively. Visually, imagine a long ribbon being cut into sections based on the number of blocks in a bar model.

Proportion: A proportion is an equation stating that two ratios are equal, such as $a:b = c:d$ or $\frac{a}{b} = \frac{c}{d}$. If two quantities are in direct proportion, as one increases, the other increases at a constant rate. This constant rate is represented visually by the slope of a straight line on a graph.

The Unitary Method: This technique involves finding the value of a single unit (the unit rate) to solve problems. For example, if $5$ pens cost $\$10$, you find the cost of $1$ pen (\$10 \div 5 = \2$) before calculating the cost of$8$pens ($8 \times $2 = $16$).

Map Scales and Scale Drawings: A scale is a ratio that compares the dimensions of a model or map to the actual size of the object. A scale of $1:100$ means $1$ unit on paper represents $100$ units in real life. Visually, a scale drawing looks like the original object but is resized (enlarged or reduced) while maintaining the same proportions.

Percentages as Ratios: A percentage is a specific type of ratio where the second quantity is always $100$. For example, $25\%$ is the ratio $25:100$, which simplifies to $1:4$. Visually, this can be seen as $25$ squares shaded in a $10 \times 10$ grid of $100$ small squares.