Ratio and Proportion - Unitary Method

The Unitary Method Definition: This is a mathematical technique used to solve problems by first finding the value of a single unit (one item) and then using that value to calculate the total value of the required number of units.

Calculating the Unit Value (Division): To find the value of one unit when the value of many is known, we divide the total value by the number of units. This is expressed as $\text{Value of 1 unit} = \\frac{\text{Total Value}}{\text{Total Quantity}}$. Visually, this corresponds to dividing a single whole into equal partitioned segments.

Calculating the Required Value (Multiplication): Once the value of one unit is known, we multiply it by the desired quantity to find the final answer. This is expressed as $\text{Total Value} = \text{Value of 1 unit} \times \text{Required Quantity}$.

Direct Variation (Direct Proportion): This occurs when an increase in one quantity leads to a proportional increase in the other (e.g., more items cost more money). On a graph, this relationship is represented by a straight line that passes through the origin $(0,0)$ and rises at a constant angle.

Inverse Variation (Inverse Proportion): This occurs when an increase in one quantity results in a proportional decrease in the other (e.g., more workers take less time to complete a task). Visually, this is represented by a downward-sloping curve that never touches the $x$ or $y$ axes.

Unit Consistency: Before performing calculations, all quantities must be in the same units. For example, if comparing weights, convert all values to either grams or kilograms using the conversion $1 \text{ kg} = 1000 \text{ g}$.

Ratio Comparison: The unitary method is often used to compare ratios. By finding the unit price or unit rate for two different deals, you can determine which one offers better value for money.