A1.1 Managing the hard middle
Every real problem has a hard middle — the point where the first idea has run out. Strategies for that moment: write down what you know, simplify the numbers to see the structure, work a small case, or explain the problem to someone — even to yourself.
Example
Solve: A store marks a $48 jacket up 25%, then offers 25% off. Is the final price $48?
- Simplify to see the structure: try $100 first. Up 25% → 125; then 25% off 125 → 93.75. Not 100.
- Now the real numbers: 48 × 1.25 = 60; 60 × 0.75 = 45. Final price $45.
Tip: The percents act on different bases — that is the insight the small case revealed.
Remember: A fast estimate you can check beats a slow answer you can't.