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 |
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Worked Examples
### Example 1 — Ratio and Proportion *The ratio of boys to girls in a class is 4 : 5. If there are 36 boys, find the number of girls.*
**Solution** Let boys = 4x, girls = 5x. Given 4x = 36 → x = 9. Girls = 5 × 9 = **45**.
### Example 4 — Successive Percentage Change *A population increases by 10% in the first year and decreases by 10% in the second year. What is the net percentage change?*
**Solution** Use net effect formula: 10 + (−10) + (10 × −10)/100 = 0 − 1 = **−1%** (a decrease of 1%).
Step-by-step check: Start with 100 → after +10% = 110 → after −10% of 110 = 110 − 11 = 99. Change = −1%.
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Common Mistakes
1. **Adding percentages directly in successive changes** Wrong: 10% up then 10% down = 0% change. Correct: The second percentage acts on the new base, not the original. Use stepwise calculation or the net-effect formula (result here is −1%).
2. **Ignoring units before forming a ratio** Wrong: Comparing 2 kg and 500 g as 2 : 500. Correct: Convert to the same unit first → 2000 g : 500 g = 4 : 1.
3. **Confusing part-to-part with part-to-whole** Wrong: Ratio 3 : 2 means 3/2 of the whole. Correct: Total parts = 5; first quantity is 3/5 of the whole, not 3/2.
4. **Reversing the unitary method in inverse variation** Wrong: More workers → more days. Correct: More workers → fewer days. Use inverse relation: days = (original workers × original days) / new workers.
5. **Forgetting to convert percent back to a usable form** Wrong: Writing 25% as 25 in multiplication. Correct: Always use 25/100 or 0.25 when computing.
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Quick Reference
Ratio a : b → fraction form a/b; always simplify by HCF.
Proportion rule: product of extremes = product of means (a × d = b × c).
A class has 18 boys and 12 girls. What is the ratio of boys to girls in simplest form?
Q2 · Ratio, Proportion and Percentage · EASY
If 5 pens cost Rs 75, what is the cost of 8 such pens using the unitary method?
Q3 · Ratio, Proportion and Percentage · MEDIUM
Two numbers are in the ratio 5 : 7. If their sum is 144, what is the larger number?
Q4 · Ratio, Proportion and Percentage · MEDIUM
A student scored 432 marks out of 600 in an examination. What is the percentage obtained by the student?
Q5 · Ratio, Proportion and Percentage · HARD
The ratio of milk to water in a mixture is 7 : 3. If the mixture contains 35 litres of milk, how much water should be added so that the ratio becomes 7 : 5?