Follow growth one period at a time
Why do repeated percentages differ from repeated fixed additions?
Entering midway? Revisit Allocate a constrained budget for the preceding worked example, or start here and use the explanation below.
Start with a concrete question
An invented stock begins at a chosen number of units and changes by a constant annual percentage. Compare compounding with changes computed from the original base.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
What do you expect?
Predict whether +10% followed by −10% returns to the starting amount.
Change one thing. Inspect what happens.
- Compare 10% growth after one, two and five years.
- Try −10% and inspect whether the absolute decrement stays constant.
- Calculate +10% then −10% manually and explain why they do not cancel.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Compare the current results
Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.Follow the calculation
- Year 1: 50 × 1.1 = 55 units.
- Year 2: 55 × 1.1 = 60.5 units.
- Year 3: 60.5 × 1.1 = 66.55 units.
- Year 4: 66.55 × 1.1 = 73.205 units.
- Year 5: 73.205 × 1.1 = 80.5255 units.
Percentage change acts on the current amount. The fixed-base comparison applies every change to the original amount and can become physically meaningless below zero.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Follow growth one period at a time
An invented stock begins at a chosen number of units and changes by a constant annual percentage. Compare compounding with changes computed from the original base.
Predict before changing inputs
Predict whether +10% followed by −10% returns to the starting amount.
Understand the mechanism
A percentage needs a base. Compounding updates the base after every period.
Starting at 50, two years of 10% growth give 50 × 1.1 × 1.1 = 60.5, compared with 60 from two fixed additions of 5.
Growth is multiplicative: +10% then −10% multiplies by 1.1 × 0.9 = 0.99.
Calculate it yourself
Stock after t periods = initial × (1 + rate)^t. Rate is expressed as a fraction.
Guided investigation
Compare 10% growth after one, two and five years.
Try −10% and inspect whether the absolute decrement stays constant.
Calculate +10% then −10% manually and explain why they do not cancel.
Assumptions and limitations
Constant growth cannot continue indefinitely in resource-limited systems.
The fixed-base comparison may cross zero for large negative totals and then ceases to describe a physical stock.
Further investigations
High school: Make a table comparing doubling with adding a fixed amount.
College: Derive doubling time using logarithms and test discrete versus continuous growth.
Working professionals: Build a scenario forecast with changing growth and capacity limits.
Try a fresh case
A stock of 80 grows 25% each period for two periods. What is the final amount?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Make a table comparing doubling with adding a fixed amount.
College investigation
Derive doubling time using logarithms and test discrete versus continuous growth.
Workplace investigation
Build a scenario forecast with changing growth and capacity limits.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.