Mathematics: Class 8 (Part 1) — Chapter 7: Proportional Reasoning-1
7.1 Observing Similarity in Change & Ratios
Concept of Proportionality
When scaling digital images, if both width and height change by the same multiplicative factor, the images look similar and are said to be proportional.
If they change by different factors (or by equal additive/subtractive amounts), the image becomes distorted.
Ratios
A ratio represents a proportional relationship between two quantities, written as $a : b$.
The numbers $a$ and $b$ are called the terms of the ratio.
For every $a$ units of the first quantity, there are $b$ units of the second quantity.
7.2 Ratios in Simplest Form & Proportion
Simplest Form of a Ratio
To reduce a ratio to its simplest form, divide both terms by their Highest Common Factor (HCF).
Example: The ratio $60 : 40$ has an HCF of $20$. Dividing both terms by $20$ yields the simplest form $3 : 2$.
Proportion
When two ratios are equal in their simplest forms, they are in proportion.
This is denoted by the symbol :: (e.g., $a : b :: c : d$, meaning $a : b$ is proportional to $c : d$).
Cross-Multiplication Rule: If $a : b :: c : d$, then $ad = bc$.
Any of the four terms can be found if the other three are known: $d = \frac{bc}{a}$.
7.3 Trairasika — The Rule of Three
Historical Context (Aryabhata)
In ancient Indian mathematics, problems of proportionality were solved using the Rule of Three (Trairasika).
It involves three known terms to find a fourth unknown term: