Ratio and Proportion

Overview

Ratio and Proportion forms a foundational pillar of primary mathematics and appears consistently in Bihar TET Paper I. This topic tests your ability to compare quantities, establish relationships between numbers, and solve real-world problems involving distribution, scaling, and equivalence. Mastery here directly supports success in related areas like percentage, profit-loss, and time-work problems.

For Bihar TET, expect 2-4 questions covering direct comparison of ratios, finding missing terms in proportions, and word problems using the unitary method. The questions typically involve simple numbers suitable for Class I-V level but test conceptual clarity. Students must understand not just the mechanical procedures but also the reasoning behind why ratios work—essential for answering pedagogy-linked questions.

Key Concepts

• **Ratio** expresses how many times one quantity contains another. The ratio of a to b is written as a:b or a/b, where b ≠ 0.

• **Antecedent and Consequent**: In the ratio a:b, 'a' is the antecedent (first term) and 'b' is the consequent (second term).

• **Equivalent Ratios**: Ratios that represent the same relationship. Multiplying or dividing both terms by the same non-zero number gives equivalent ratios (2:3 = 4:6 = 6:9).

• **Simplest Form**: A ratio is in simplest form when the HCF of both terms is 1. Always reduce ratios to simplest form before comparing.

• **Proportion** states that two ratios are equal. If a:b = c:d, then a, b, c, d are in proportion, written as a:b :: c:d.

• **Mean Proportion**: If a:b = b:c, then b is the mean proportional between a and c, and b = √(a×c).

• **Unitary Method**: A technique to find the value of multiple units by first finding the value of one unit. Foundation of direct and inverse variation.

• **Direct Proportion**: When two quantities increase or decrease together at the same rate (more items cost more money).

Formulas / Key Facts

| Concept | Formula/Rule | |---------|--------------| | Ratio of a to b | a:b = a/b | | Product of extremes = Product of means | If a:b :: c:d, then a×d = b×c | | Finding fourth proportional | If a:b :: c:x, then x = (b×c)/a | | Finding third proportional | If a:b :: b:x, then x = b²/a | | Mean proportional of a and c | b = √(a×c) | | Dividing N in ratio a:b | Parts are Na/(a+b) and Nb/(a+b) | | Comparison of ratios | Convert to fractions with same denominator, or cross-multiply | | Unitary method (direct) | If 1 unit = value, then n units = n × value |

**Must-Remember Facts:**