Ratio and proportion — Question VIS-02
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This question tests your understanding of the following concepts from the chapter Ratio and proportion: Second, Ingestion, Test, Published, Ratio and proportion, Mathematics. These are fundamental topics in Mathematics that students are expected to master as part of the ICSE Class 10 curriculum.
A thorough understanding of these concepts will help you answer similar questions confidently in your ICSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This Question
Read the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to Remember
- Always show your working steps clearly.
- Verify your answer by substituting values back into the equation.
- Practice similar problems from the textbook exercises.
- Memorise important formulae and their conditions of applicability.
Practice more questions from Ratio and proportion — Mathematics, Class 10 ICSE
Chapter Overview: Ratio and Proportion
This chapter extends the concepts of ratio and proportion learned in earlier classes. Students work with componendo-dividendo, properties of equal ratios (alternendo, invertendo, componendo, dividendo), and problems involving continued proportion. The chapter is essential for solving complex algebraic manipulations involving ratios.
Key problem types include: proving identities using properties of proportion, finding unknown values when given a proportion, and applying componendo-dividendo to simplify expressions. These skills are also useful in other chapters like similarity and trigonometry.
Board Exam Weightage: 4-5 marks | Difficulty: Moderate
Key Properties and Formulas
| Property | If a/b = c/d then |
|---|---|
| Invertendo | b/a = d/c |
| Alternendo | a/c = b/d |
| Componendo | (a+b)/b = (c+d)/d |
| Dividendo | (a−b)/b = (c−d)/d |
| Componendo-Dividendo | (a+b)/(a−b) = (c+d)/(c−d) |
| Continued Proportion | a/b = b/c → b² = ac (b is the mean proportional) |
Must-Know Concepts
- Let a/b = c/d = k: Then a = bk, c = dk — substitute into expressions to prove identities
- Componendo-Dividendo: The most powerful technique — use when expression has (a+b)/(a−b) form
- Mean proportional: If a, b, c are in continued proportion, b = √(ac)
- Third proportional: If a : b = b : c, then c = b²/a
- Cross multiplication: If a/b = c/d, then ad = bc
Common Mistakes to Avoid
- Confusing componendo (adding 1) with dividendo (subtracting 1)
- Applying componendo-dividendo when a/b ≠ c/d — the property only works for equal ratios
- Forgetting to substitute back after letting ratio = k
- Errors in algebraic simplification after applying properties
Scoring Tips
- For "prove that" questions, start from LHS and simplify to get RHS (or vice versa)
- The k-method (let ratio = k) is the most reliable approach for complex proofs
- For componendo-dividendo, clearly show the steps: apply componendo, apply dividendo, then divide
- Always simplify your final answer to the simplest form
Frequently Asked Questions
What is the k-method?
If a/b = c/d, let each ratio equal k. Then a = bk and c = dk. Substitute these into the expression you need to prove or simplify. This converts the problem into algebra with a single variable k.
When should I use componendo-dividendo?
Use it when you see expressions of the form (a+b)/(a−b) or when you need to eliminate fractions. It is especially useful when the given ratio is complex and direct substitution is cumbersome.
What is the difference between mean proportional and third proportional?
Mean proportional of a and c is b where a/b = b/c, so b = √(ac). Third proportional to a and b is c where a/b = b/c, so c = b²/a. Mean proportional is between two numbers; third proportional follows two numbers.