Ratio and Proportion - Unitary Method

Definition of Ratio: A ratio is a comparison of two quantities of the same kind and in the same units by division, expressed as $a:b$ or $\frac{a}{b}$. Visually, imagine two line segments where one is twice as long as the other; the ratio of their lengths is $2:1$, regardless of whether they are measured in cm or inches.

Simplest Form of Ratio: A ratio is in its simplest form when the antecedent (first term) and the consequent (second term) have no common factor other than $1$. Visualize a large rectangle divided into $12$ small squares with $4$ colored red; the ratio $4:12$ can be simplified to $1:3$, meaning for every $1$ red square, there are $3$ total squares.

Concept of Proportion: An equality of two ratios is called a proportion, written as $a:b = c:d$ or $a:b :: c:d$. Imagine a photograph being enlarged; to keep the image from looking distorted, the ratio of height to width in the original must equal the ratio of height to width in the enlargement.

Terms of Proportion: In the proportion $a:b :: c:d$, $a$ and $d$ are called the 'extremes' while $b$ and $c$ are called the 'means'. A helpful visual is to see them as a chain where the outer links are the extremes and the inner links are the means, and for the proportion to hold, the product of the outer links must equal the product of the inner links.

The Unitary Method: This is a technique used to solve problems by first finding the value of a single unit and then finding the necessary value by multiplying the single unit value. Think of a carton containing $12$ eggs; to find the cost of $5$ eggs, you first 'zoom in' to find the price of just $1$ egg.

Finding Value of One Unit: To find the value of one unit, we use division. The rule is: $\text{Value of one unit} = \frac{\text{Total value}}{\text{Total quantity}}$. Visualize sharing $20$ candies equally among $5$ children to determine how many candies $1$ child receives.

Finding Value of Multiple Units: Once the value of $1$ unit is known, we use multiplication to find the value of the required number of units. $\text{Value of } n ext{ units} = \text{Value of } 1 ext{ unit} \times n$. Imagine knowing the weight of $1$ brick and stacking $10$ of them to find the total weight.

Direct Variation: In standard unitary method problems for Grade 6, quantities usually increase or decrease together. This can be visualized as a straight-line graph passing through the origin $(0,0)$; if you buy $0$ items, you pay $0$ amount, and as you move right on the quantity axis, the cost axis goes up at a steady rate.