These questions apply two percentage changes in sequence and ask for a final value or an overall change. Each stage has a new base, so the reliable method is to build and multiply a chain of multipliers.
Build the multiplier chain
Read the changes in order. Replace a rise of p% by 1 + p/100 and a fall of p% by 1 - p/100, then multiply the starting value by every multiplier. For example, a starting value of 250 followed by a 12% rise and a 5% fall becomes 250 × 1.12 × 0.95 = 266. The second change acts on 280, the value after the first stage, not on 250.
To find the overall percentage change without a starting value, multiply only the change multipliers. In the example, 1.12 × 0.95 = 1.064. Because the product is 0.064 above 1, the overall result is a 6.4% increase. A product below 1 represents a decrease; subtract it from 1 and convert the difference to a percentage. A product equal to 1 means the changes exactly restore the start.
| Signal | Action |
|---|---|
| Rise by p% | Multiply by 1 + p/100 |
| Fall by p% | Multiply by 1 - p/100 |
| Several changes | Multiply all stage multipliers |
| Overall change requested | Compare the multiplier product with 1 |
The traps
| Tempting route | How to reject it |
|---|---|
| Add or subtract the stated rates | That keeps the starting value as the base for both stages. The later rate acts on an updated value, so multiply the stage multipliers instead. |
| Assume equal opposite rates cancel | A fall after a rise is taken from a different base. Check the product of the two multipliers; equal numerical rates usually leave a product below 1. |
| Stop after the first change or apply only the second | A successive-change result must include every stated stage. Write one multiplier per stage before doing any arithmetic. |
| Read a multiplier of 1.50 as a 150% increase | The whole starting value accounts for 100%. Only the amount above 1 is the increase, so compare the product with 1. |
Try it
A workshop makes 200 parts a day. Output grows by 10% in one month and by another 10% the next month. Enter the final number of parts per day.
A soup recipe uses 80 millilitres of cream. A chef increases it by 25%, then reduces the new amount by 20%. Enter the final number of millilitres.
A bus route carries 1000 riders. Ridership increases by 8%, then the updated number increases by 5%. Enter the final number of riders.