Ratio & Proportion: Advanced techniques for solving complex business problems using continued proportions, mean proportionals, and compound ratios.
## Core concept
Advanced ratio and proportion builds on basic definitions to solve multi-step business scenarios. At CA Foundation level, you must master:
- Continued proportion: Three or more quantities in succession where a:b = b:c = c:d. Here, b and c are called *mean proportionals*.
- Compound ratio: Combination of two or more ratios by multiplying corresponding terms.
- Applications: Problems involving mixtures, profit sharing, resource allocation, and inverse relationships.
Unlike basic ratio (comparing two quantities), advanced topics require systematic manipulation of multiple ratios simultaneously and understanding when to use multiplication vs. inversion.
## Formulas & rules
Continued proportion (a, b, c in continued proportion) - Condition: a:b = b:c - Derived: b² = a × c (geometric mean property) - Solve for missing term: b = √(a × c)
Compound ratio - If a:b and c:d are two ratios, compound ratio = (a × c):(b × d) - For n ratios: multiply all antecedents and all consequents separately
Properties - If a:b = c:d and e:f = g:h, then (a×c×e):(b×d×f) = compound of all three ratios - Inverse ratio of a:b is b:a - Duplicate ratio of a:b is a²:b² - Subduplicate ratio of a:b is √a:√b
Solving word problems 1. Identify all ratios from the problem statement 2. Determine whether to use compound, continued, or inverse ratio 3. Set up the equation using the appropriate ratio relationship 4. Solve for the unknown(s)
## Worked example
Problem: Three quantities are in continued proportion. The first is 4 and the third is 16. Find the second quantity.
Solution: - Let the quantities be a, b, c where a = 4, c = 16 - Continued proportion condition: a:b = b:c - Therefore: 4:b = b:16 - Cross-multiply: b² = 4 × 16 = 64 - b = 8
Verification: 4:8 = 8:16 → 1:2 = 1:2 ✓
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Problem: In a business partnership, profits are to be divided in the ratio 3:5:7. If the total profit is ₹3,000, find each partner's share using compound ratio principles.
Solution: - Total parts = 3 + 5 + 7 = 15 - First partner: (3/15) × 3,000 = ₹600 - Second partner: (5/15) × 3,000 = ₹1,000 - Third partner: (7/15) × 3,000 = ₹1,400
## Common exam applications
- Mixture problems: Combining liquids in different ratios; finding resultant concentration
- Work and time: Inverse ratio relationships (more workers = less time)
- Profit sharing: Dividing profits among partners based on capital or time invested
- Speed and distance: Compound ratio in relative motion problems
- Alligation: Finding mean price/proportion when mixing items of different costs
## Common mistakes
- Forgetting to square in continued proportion: Writing b = a × c instead of b = √(a × c)
- Reversing inverse ratio: Using a:b instead of b:a when the relationship is inverse
- Ignoring the "continued" condition: Treating three quantities as simple ratios rather than checking if they satisfy b² = ac
- Not simplifying before solving: Working with unsimplified ratios leads to arithmetic errors
- Confusing compound with addition: Compound ratio multiplies terms; you don't add the ratios
## Exam strategy
- Always verify continued proportion by checking if b² = a × c
- In multi-step problems, break down into simpler ratio pairs
- Double-check word problem interpretation before setting up the equation
- Practice with real business scenarios (partnership, mixture, allocation) for confidence