Ratio and Proportion

Overview

Ratio and Proportion form the backbone of quantitative reasoning at the primary level and appear consistently in OTET Paper I Mathematics. This topic tests a candidate's ability to compare quantities, establish relationships between numbers, and solve practical problems involving distribution, scaling, and equivalence.

For OTET, you must master three interconnected concepts: ratio (comparing two quantities of the same kind), proportion (equality of two ratios), and the unitary method (finding the value of one unit to calculate any required quantity). These concepts have direct classroom applications—from dividing sweets among children to calculating ingredients in recipes—making them essential for primary mathematics pedagogy.

Questions typically involve finding equivalent ratios, solving proportion problems, and applying the unitary method to real-life situations. Expect 2-4 questions from this topic, often presented as word problems requiring careful reading and systematic solving.

Key Concepts

• **Ratio** compares two quantities of the same unit and is written as a:b or a/b, where b ≠ 0. The first term (a) is called the antecedent and the second term (b) is called the consequent.

• **Ratio has no unit** because it compares like quantities. When we say the ratio of boys to girls is 3:2, we are not measuring boys or girls—we are comparing their numbers.

• **Equivalent ratios** are obtained by multiplying or dividing both terms by the same non-zero number. Thus 2:3 = 4:6 = 6:9 = 10:15.

• **Simplest form** of a ratio is obtained when the HCF of both terms is 1. To simplify 12:18, divide both by HCF (6) to get 2:3.

• **Proportion** states that two ratios are equal. If a:b = c:d, we write a:b :: c:d. Here a and d are called extremes, while b and c are called means.

• **Product of means = Product of extremes** is the fundamental property of proportion. If a:b :: c:d, then b × c = a × d.

• **Unitary method** finds the value of one unit first, then uses it to find the value of any required number of units. It relies on direct and inverse relationships.

• **Direct proportion**: When one quantity increases, the other also increases (more items cost more money). **Inverse proportion**: When one quantity increases, the other decreases (more workers take less time).

Formulas / Key Facts

| Concept | Formula/Rule | |---------|--------------| | Ratio of a to b | a:b = a/b (b ≠ 0) | | Equivalent ratio | a:b = (a×k):(b×k) for any k ≠ 0 | | Simplest form | Divide both terms by their HCF | | Proportion test | If a:b :: c:d, then a×d = b×c | | Mean proportional | If a:x :: x:b, then x = √(a×b) | | Dividing in ratio | If amount P is divided in ratio a:b, parts are Pa/(a+b) and Pb/(a+b) | | Unitary method (direct) | Value of n units = (Value of given units ÷ given units) × n | | Unitary method (inverse) | If 5 workers take 12 days, 1 worker takes 5×12 = 60 days |