Level 7: Multi-Step Analysis and Reasonable-Answer Judgment
Level 7 questions are the most demanding: several steps, mixed operations, and often a request to judge whether an answer is reasonable or to choose the best plan. The math is within reach; the work is in organizing it.
Concepts to know
- Chained operations: cost plus tax plus shipping; a discount followed by a coupon; a rate applied over a variable time.
- Break-even and comparison: total cost = fixed cost + (rate × quantity); set two plans equal to compare.
- Estimation and reasonableness: round to convenient numbers to sanity-check an exact answer.
- Weighted averages: multiply each value by its weight, sum, then divide by total weight.
- Backward solving: work from the final result to the starting value using inverse operations.
Traps to watch
- Applying a percent to the wrong base. A discount applies to the price, a tax applies to the discounted price.
- Comparing totals, not rates. A lower per-unit rate can still cost more if a minimum or fee applies.
- Ignoring constraints. Questions may specify a budget cap or a time limit; the correct answer must satisfy it.
- Rounding too early. Carry decimals through intermediate steps and round only at the end.
- Plausibility. If a monthly bill computes to $4,000 for a single phone line, recheck the setup.
Rule of thumb
Write the problem as a chain of labeled steps before computing: Step 1 → Step 2 → Step 3. At the end, estimate the answer roughly; if the exact result is far from the estimate, find the error.
Worked example
Plan A charges $30 plus $0.10 per minute. Plan B charges $50 plus $0.05 per minute. For how many minutes do they cost the same? Set 30 + 0.10m = 50 + 0.05m. Then 0.05m = 20, so m = 400 minutes. Beyond 400 minutes, Plan B is cheaper.