Ratio, Proportion and Percentage

Overview

Ratio, proportion and percentage form the quantitative backbone of primary and upper-primary mathematics. These concepts connect pure arithmetic to real-world problem-solving—comparing quantities, scaling recipes, calculating discounts, understanding maps and models, and interpreting data. For UPTET, this topic appears both in the content section (direct calculation questions) and in pedagogy-linked questions where you must identify student misconceptions or suggest teaching strategies.

Mastery here means fluency in three interconnected ideas: ratio (comparing two quantities of the same kind), proportion (equality of two ratios), and percentage (a ratio with denominator 100). The unitary method—finding the value of one unit first—is the universal problem-solving tool that ties them together. Expect 3–5 questions directly from this cluster, often framed as word problems involving money, time, distance, mixtures or simple data interpretation.

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Key Concepts

• **Ratio** expresses how many times one quantity contains another. Written as a : b or a/b, both quantities must be in the same unit. A ratio has no unit itself.

• **Equivalent ratios** are obtained by multiplying or dividing both terms by the same non-zero number (e.g., 2 : 3 = 4 : 6 = 6 : 9).

• **Proportion** states that two ratios are equal: a : b :: c : d (read "a is to b as c is to d"). The cross-product rule holds: a × d = b × c.

• **Unitary method** finds the value of one unit first, then scales to the required number of units. It works for direct variation (more → more) and inverse variation (more → less).

• **Percentage** means "per hundred." To convert a fraction to percent, multiply by 100; to convert percent to fraction, divide by 100.

• **Percentage change** = (Change / Original) × 100. Increase uses a positive change; decrease uses a negative change.

• **Successive percentage changes** do not simply add; use the formula or calculate stepwise on the new base each time.

• **Part-to-whole vs part-to-part**: Ratio 2 : 3 can mean 2 parts out of 5 (part-to-whole = 2/5) or comparison of two parts directly.

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Ratio simplification | Divide both terms by their HCF | | Proportion cross-product | If a : b :: c : d, then a × d = b × c | | Fraction to percent | (Fraction) × 100 | | Percent to fraction | (Percent) / 100 | | Percentage of a number | (Percent / 100) × Number | | Percentage increase | New = Original × (1 + r/100) | | Percentage decrease | New = Original × (1 − r/100) | | Finding original after increase | Original = New / (1 + r/100) | | Successive changes (r₁%, r₂%) | Net effect = r₁ + r₂ + (r₁ × r₂)/100 (use signs) | | Unitary method (direct) | If M₁ items cost C₁, then M₂ items cost (C₁/M₁) × M₂ | | Unitary method (inverse) | If M₁ workers finish in D₁ days, M₂ workers finish in (M₁ × D₁)/M₂ days |