Problem Solving
About 6 min · Achievers Section · English
Time worked out from the lesson's 407 words at 150 words a minute, plus a minute for each question.
The Problem Solving section in the SOF IMO Achievers segment tests your ability to synthesize multiple mathematical concepts and logical reasoning skills into a single solution pathway. Unlike standard questions that focus on one topic, these problems deliberately blend arithmetic, algebra, geometry, number theory, and logical reasoning. Success here demonstrates mathematical maturity — the capacity to recognize patterns, choose appropriate tools, and execute multi-step solutions under time pressure.
Key ideas
- Cross-topic integration: Problems intentionally combine 2–4 syllabus areas (e.g., coordinate geometry + arithmetic progressions, or circles + number patterns). You must recognize which tools apply to which part of the problem.
- Multi-step reasoning: Solutions typically require 3–5 logical steps. Each step uses the output of the previous one. Missing or miscomputing one intermediate value cascades into a wrong final answer.
- Real-world contexts: Many problems are framed as practical scenarios (construction layouts, race timings, mixture problems, price optimization) requiring you to mathematically model the situation before solving.
- Strategic guessing points: When stuck, identify which intermediate result you can verify independently. Sometimes working backward from answer options reveals the solution pathway.
- Time management: If you're not making progress after 2 minutes, mark for review and move on. Returning with fresh eyes often helps.
Quick reference
- Mixed problems = Identify components first, solve each, combine results.
- Draw diagrams for any spatial relationships — visualization prevents algebraic errors.
- When stuck, try: change perspective, work backward, test extreme cases, simplify the problem.
- Common bridges: percent ↔ fraction ↔ decimal ↔ ratio — convert to the most convenient form.
- Time budget: 3–4 minutes per problem; if no progress in 2 minutes, mark and return later.
Check yourself: 3 questions
1. A rectangular park has length 80 m and breadth 60 m. A path of uniform width runs all around inside the park. If the area of the path is 1056 sq m, what is the width of the path?
2. In a class, 60% of students like cricket, 50% like football and 30% like both. If 10 students like neither cricket nor football, how many students are there in the class?
3. A tank can be filled by two pipes A and B in 12 hours and 18 hours respectively. A third pipe C can empty the full tank in 9 hours. If all three pipes are opened together when the tank is empty, how long will it take to fill the tank completely?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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