Decision and conditions
Distinguish final balances, profits, margins and revenue multipliers. Profit margin divides by selling price; markup divides by cost. Compounding uses the changing balance.
Worked example
$300 is invested for two years. What is the excess interest earned at 10% compounded annually over 10% simple annual interest? Compounded interest is 0.21(300)=63. Simple interest is 0.20(300)=60. The difference is $3. Method choice: Only interest on first-year interest creates the excess in year two.
Contrast: change the reasoning
A price is increased by 20%, then discounted by 25%, leaving $270. What was its original price? The net multiplier is 1.2×0.75=0.9. 0.9p=270. The original price is $300. Method choice: Combine the successive multipliers and then reverse them.
Transfer with solution
An item costs $180. Profit must be 25% of selling price. What selling price is required? Let price be p; p−180=p/4. Thus 3p/4=180. p=240. Method choice: Selling price minus cost equals one quarter of selling price, not one quarter of cost.
Boundary check
A product’s price rises by 30% and quantity sold falls by 20%. What is the new revenue as a percent of the old revenue? The price multiplier is 13/10. The quantity multiplier is 0.8. New revenue as a percent is 104. Method choice: Revenue is price times quantity, so combine their multipliers.
Misconception to reject
A price increase does not by itself imply an equal revenue increase. Reports all compound interest rather than its excess over simple interest.
$300 is invested for two years. What is the excess interest earned at 10% compounded annually over 10% simple annual interest?
- Compounded interest is 0.21(300)=63.
- Simple interest is 0.20(300)=60.
- The difference is $3.
Compare the other answer choices
Reports one year’s interest, not interest on that interest.
Reports all simple interest.
Reports all compound interest rather than its excess over simple interest.
Reports the final compounded balance including principal.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
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