Simple interest, compound growth and decay · Higher
| English | Português |
|---|---|
| compound interest/ˈkɒmpaʊnd ˈɪntrest/ | juros compostos |
A 1000-yuan deposit earns 10% a year. Simple interest adds the same 100 yuan each year; compound interest earns interest on earlier interest too.
- A 1000-yuan deposit earns 10% a year. Simple interest adds the same 100 yuan each year; compound interest earns interest on earlier interest too.
- This lesson studies compound interest 复利: Interest calculated on the updated balance each period.
Choose the mathematical structure
- A percentage r corresponds to decimal r/100; increases use multiplier 1+r/100 and decreases 1-r/100. Simple interest on principal P for n periods at rate r is Prn/100, giving balance P(1+rn/100). Compound balance is P(1+r/100)^n. Decay uses the corresponding decreasing multiplier. Use matching rate and period units, and divide by the multiplier to recover an original value.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines compound interest?
Interest calculated on the updated balance each period.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
At 10% per year, 1000 yuan earns simple interest 100 per year: after two years interest is 200 and balance is 1200. Compound balances are 1100 after year one and 1210 after year two. A machine worth 800 yuan losing 20% each year becomes 640 then 512, not 480: the second loss is 20% of 640. An 800-yuan sale price after a 20% reduction corresponds to original price 800/0.8=1000. Growth from 50 to 65 is (65-50)/50×100=30%; 65 is 130% of 50. If a compound balance must first exceed 1300 at 10%, year two gives 1210 and year three 1331, so three whole years are needed.
Simple interest, compound growth and decay
A percentage r corresponds to decimal r/100; increases use multiplier 1+r/100 and decreases 1-r/100
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find a 1000-yuan balance after two years at 10% compound annual interest.
1000×1.1²=1210.
Test a tempting shortcut
- Use the original value as the denominator for percentage change. Repeated 20% decreases do not subtract 40% of the initial amount. A rate per year cannot be treated as a rate per month without a specified conversion.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Simple and compound interest always give the same balance after two years. This claim is false. Explain which definition or assumption it violates.
Find an 800-yuan value after two annual 20% reductions.
800×0.8²=512.
Simple and compound interest always give the same balance after two years.
Use the original value as the denominator for percentage change. Repeated 20% decreases do not subtract 40% of the initial amount. A rate per year cannot be treated as a rate per month without a specified conversion.
Interpret a new situation
- AQA R9/R16 includes percentage comparisons, original values, simple interest and repeated growth/decay. State whether the task asks for interest alone or total balance, and interpret whole-period threshold answers.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find total simple interest on 1000 yuan at 10% for two years.
1000×0.10×2=200; interest is separate from balance.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
Interest calculated on the updated balance each period. Choose the relationship, show the method, check its assumptions and interpret the result.