REVISION SUMMARY: Proportional Reasoning-1 (Chapter 7)
1. Chapter at a glance
- Images appear similar when their widths and heights change by the same multiplication factor (proportional change); additive changes (same difference) distort the shape.
- A ratio a : b expresses “for every a units of the first quantity there are b units of the second”; terms are reduced to simplest form by dividing by their HCF.
- Two ratios a : b and c : d are proportional (written a : b :: c : d) when their simplest forms are identical or when ad = bc.
- The Rule of Three (Āryabhaṭa) solves a : b :: c : d by cross-multiplication: d = (b × c) / a; units of corresponding quantities must be the same.
- To divide a quantity x in the ratio m : n, the parts are m × x / (m + n) and n × x / (m + n).
- Real-life problems (lemonade sweetness, wall strength, paint mixtures, manure quantities) are solved by checking or setting up proportional ratios.
- Unit conversion is essential before applying proportionality (minutes ↔ hours, grams ↔ kilograms, sq ft ↔ acres).
2. Definitions, theorems and results
- Ratio: In a : b, for every a units of the first quantity there are b units of the second; a and b are the terms.
- Simplest form of a ratio: obtained by dividing both terms by their HCF.
- Proportional ratios: a : b :: c : d means the ratios are equal in simplest form (or ad = bc). The symbol :: indicates proportionality.
- Cross-multiplication result (Rule of Three): If a : b :: c : d then ad = bc; the unknown fourth term is found by d = (b × c) / a. (No formal proof required in exam; derived from equal factors of change.)
- Division in a given ratio: When x is divided in ratio m : n the parts are m × x / (m + n) and n × x / (m + n). (Derived directly from grouping into m + n equal parts.)
3. Formula sheet
| Formula |
Meaning of symbols |
| a : b :: c : d ⇔ ad = bc |
a, b, c, d are terms of two ratios; equality holds when ratios are proportional |
| d = (b × c) / a |
Fourth proportional when a : b :: c : d |
Part 1 = m × x / (m + n)
Part 2 = n × x / (m + n) |
x divided in ratio m : n; m + n is total parts |
| Factor of change f = c / a = d / b |
Same multiplier applied to both terms of a ratio |
4. Solved-example patterns
- Check if two ratios are proportional: reduce both to simplest form (divide by HCF) and compare; alternatively verify ad = bc.
- Find missing term in a : b :: c : ? : compute factor f = c / a, then multiply b by f; or use cross-multiplication d = (b × c) / a (ensure identical units).
- Divide a quantity x in ratio m : n: compute total parts m + n, size of one part = x / (m + n), then multiply.
- Real-life proportional situations (sweetness, strength, mixtures): set up ratio of two related quantities, form a : b :: c : ?, solve with same factor or cross-multiplication.
- Compare costs or densities after converting to common units (e.g., price per kg).
5. Common mistakes and exam pitfalls
- Using subtraction instead of multiplication factor when checking similarity (e.g., “both dimensions decreased by 20 mm”).
- Forgetting to convert units before setting up proportion (minutes vs hours, grams vs kg, sq ft vs acres).
- Writing the proportion in wrong order (mixing “teachers : students” with “students : teachers”).
- Assuming every pair of ratios is proportional without verifying simplest forms or ad = bc.
- Incorrect grouping when dividing in ratio (forgetting to add m + n to find total parts).
- Leaving answers without simplest-form ratios or without stating units.