Ratio and Proportion - Proportion

Definition of Proportion: A proportion is an equality between two ratios. If two ratios $\frac{a}{b}$ and $\frac{c}{d}$ are equal, we say the four numbers $a, b, c, d$ are in proportion. Visually, imagine two identical shapes of different sizes; if the ratio of their width to height remains constant, they are proportional.

Notations Used: Proportion is represented using the symbol $::$ or $=$. For example, $a : b :: c : d$ is read as '$a$ is to $b$ as $c$ is to $d$'. You can visualize this as two fraction bars separated by an equal sign: $\frac{a}{b} = \frac{c}{d}$.

Terms of Proportion: In the proportion $a : b :: c : d$, the four numbers involved are called 'terms'. $a$ is the first term, $b$ is the second, $c$ is the third, and $d$ is the fourth. They must be considered in that specific order.

Extremes and Means: In the sequence $a, b, c, d$, the first and fourth terms ($a$ and $d$) are called the Extremes (outer terms), while the second and third terms ($b$ and $c$) are called the Means (middle terms). You can visualize this by drawing a line connecting the outermost numbers and another line connecting the innermost numbers.

The Equality Property: For four numbers to be in proportion, the product of the extremes must equal the product of the means ($a \times d = b \times c$). If you draw an 'X' across the proportion $\frac{a}{b} = \frac{c}{d}$, the lines of the 'X' show which numbers should be multiplied together to check for equality.

Checking for Proportion: To verify if two ratios form a proportion, simplify both ratios to their lowest terms. If both simplified fractions are identical, the ratios are in proportion. For example, $2 : 4$ and $10 : 20$ both simplify to $1 : 2$, showing they are proportional.

Order Sensitivity: The order of terms is crucial. If $a, b, c, d$ are in proportion, it does not automatically mean $a, c, b, d$ are in proportion. Changing the position of the numbers changes the ratios they form.