Ratio and Proportion - Proportion as Equality of Ratios

Definition of Proportion: A proportion is an equation that states that two ratios are equal. If the ratio $a:b$ is equal to the ratio $c:d$, we say that $a, b, c,$ and $d$ are in proportion, written as $a:b = c:d$ or $a:b :: c:d$. This can be visualized as two fractions representing the same shaded part of a whole, even if the total parts differ.

Terms of Proportion: 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 known as the Extremes (outer terms), while the second and third terms ($b$ and $c$) are known as the Means (inner terms).

Product Rule of Proportion: For four numbers to be in proportion, the product of the extremes must be equal to the product of the means. Mathematically, if $a:b :: c:d$, then $a \times d = b \times c$. This is often visualized using 'cross-multiplication' where lines connect the top-left to bottom-right and top-right to bottom-left of two equal fractions.

Continued 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 ($a:b = b:c$). In this case, $b$ is called the Mean Proportional between $a$ and $c$, and $c$ is the third proportional.

Order of Terms: The order of terms in a proportion is crucial. If $a, b, c, d$ are in proportion, it does not necessarily mean $a, c, b, d$ are in proportion. You can visualize this by arranging numbers in a sequence where the 'scaling factor' must remain consistent between the first pair and the second pair.

Simplest Form Verification: One way to check if two ratios form a proportion is to reduce both ratios to their simplest form. If both simplified ratios are identical (e.g., both reduce to $2:3$), then the four numbers are in proportion. Imagine two different sized rectangles that have the same length-to-width ratio; they are proportional.

Unitary Method Application: Proportion is the basis of the unitary method. If we know that $10$ pens cost $50$ rupees, the ratio of pens to cost is $10:50$. To find the cost of $15$ pens, we find the cost of $1$ unit first and then set up a proportion: $10:50 = 15:x$.