This explorer applies percentage changes one after another and shows the base at every stage. Seeing the base move is what makes successive percentages make sense.
What does the explorer do?
You give it a starting amount and a list of percentage changes in order. For each change it works out:
- the base, meaning the amount the percentage is taken of,
- the change as a percentage,
- the multiplier (1 plus the percentage as a decimal),
- the amount after the change.
At the end it gives the net result, the overall percentage change, and the single multiplier that does the whole job in one step.
How do I use it?
- Type a starting amount. It opens with 100.
- Type the changes in order, separated by commas. Use a minus sign for a decrease, for example
20, -20, 5. - Press “Show the trace” and read the table one stage at a time.
- Change one number and run it again to see what moves. Use Reset to return to the example.
How do I read the result?
Read each row from left to right: the base, the percentage change, the multiplier, then the amount after. The amount after in one row becomes the base in the next row. That hand-over is the whole idea.
The net line says something like “Net result: 96, an overall change of −4%”. The next line gives the same result as a single multiplier, 0.96. If you compare that with 1.2 × 0.8, you see the multipliers multiply, while the percentages do not simply add.
Example walk-through
Example 1: the opening case. Start with 100 and apply +20% then −20%.
| Stage | Base | Change | Multiplier | Amount after |
|---|---|---|---|---|
| 1 | 100 | +20% | 1.2 | 120 |
| 2 | 120 | −20% | 0.8 | 96 |
The fall is 20% of 120, which is 24, not 20% of 100. So 120 − 24 = 96. The net change is −4%, and the single multiplier is 1.2 × 0.8 = 0.96.
Example 2: reverse the order. Start with 200, apply −10% then +10%. The first stage gives 200 × 0.9 = 180. The second gives 180 × 1.1 = 198. The net change is −1%. The order does not change the final amount here, because multiplication can be done in either order, but the amounts at each stage differ.
Example 3: two equal rises. Start with 100, apply +20% then +20%. Stage 1 gives 120. Stage 2 is 20% of 120, which is 24, giving 144. The net change is +44%, not +40%, because the second rise is taken of the larger base.
Assumptions and limits
- The percentages are applied in the order you enter them, each to the current amount.
- A decrease of 100% or more is not accepted.
- The numbers are fictional teaching numbers only. This is not a financial projection, and it does not model charges, tax or timing.
- Displayed values are rounded for reading. In an exam, keep exact working until the final step and follow the rounding instruction in the question.
Where to read more
The lessons behind this tool are applying successive increases and decreases and explaining why an equal percentage rise and fall do not cancel. To work backwards from a final amount, see reversing a percentage increase. Then test yourself on the percentages and changing bases practice set.
Percentages also appear in comparing exact and approximate answers, and the Mathematics learning guide shows where they fit in the course. If a method choice in mixed questions is the real difficulty, try losing the method in a mixed question. The tools page lists the others, and online one-to-one Mathematics tuition describes how our teachers work through this kind of reasoning.