Ratio and Proportion - Proportion as an equality of two ratios

A proportion is a statement that expresses the equality of two ratios. If two ratios $a:b$ and $c:d$ are equal, we say that $a, b, c, d$ are in proportion, written as $a:b :: c:d$.

In the proportion $a:b :: c:d$, the four numbers $a, b, c, d$ are called its terms. The first and fourth terms ($a$ and $d$) are called the Extremes, while the second and third terms ($b$ and $c$) are called the Means. You can visualize this as a sandwich where the Extremes are the outer crusts and the Means are the filling inside.

The fundamental property of proportion states that the Product of Extremes is always equal to the Product of Means. This is expressed as $a \times d = b \times c$. If this equality does not hold, the numbers are not in proportion.

Three numbers $a, b, c$ are said to be in Continued Proportion if the ratio of the first to the second is equal to the ratio of the second to the third, written as $a:b = b:c$. In this case, $b$ is called the Mean Proportional between $a$ and $c$.

To determine if four numbers are in proportion, you can either simplify both ratios to their lowest terms and see if they match, or use the cross-multiplication method where you visualize an 'X' shape connecting the numerator of one fraction to the denominator of the other.

The Fourth Proportional is the fourth term in a proportion. If we have $a, b, c$ and we need to find the fourth proportional $x$, we set up the equation $a:b :: c:x$ and solve for $x$ using the product rule.

In a Mean Proportional relationship $a:b :: b:c$, the term $c$ is known as the Third Proportional to $a$ and $b$. Visually, this creates a geometric sequence where each term is multiplied by the same factor to get the next.