Multiply by a fixed factor each step
In exponential growth or decay the quantity is multiplied by the same factor every period. After periods, . This is repeated proportional change — no knowledge of is needed.
Turn a percentage rate into a multiplier
For growth of per period the multiplier is ; for decay (decrease/depreciation) of it is . So gives and gives .
Use a power, don't add the % n times
Compound (exponential) change applies the multiplier times as a power, e.g. for years at . This is bigger than just adding , because each year's growth is itself grown.
Drawn from real examiner reports.
Treating exponential change as linear
Subtracting the same fixed amount each year (e.g. always -\3000$) is simple depreciation, not exponential. Exponential decay multiplies by a fixed factor, so the amount lost shrinks each year. Mixing these up is the classic error.
Using the wrong multiplier for a decrease
For a decrease the multiplier is , not . Multiplying by keeps only of the value; multiplying by correctly keeps the remaining .
Using 0.04 instead of 1.04 for growth
For a increase the multiplier is , not . Multiplying by keeps only of the amount; multiplying by correctly adds the on top of the whole.
Assuming equal % gain and loss cancel
A rise then a fall does not return the start value, and equal growth and decay rates give unequal money changes — because each percentage is taken of a different (changing) amount.
Halving once instead of using a power
"Halving each day for days" means (three times), not subtracting half once. Apply the multiplier as a power , so , not .
Read the number of periods carefully
The exponent counts the periods elapsed, not the years labelled. "The value after years" uses ; "at the start of the th year" still means complete years have passed. Count the multiplications, then raise the factor to that power.
Convert the % to a multiplier first
Turn the rate into a multiplier before you start: growth of gives and decay of gives . So is and is .
Read halves/doubles/triples as ×0.5/×2/×3
Word cues give the multiplier directly: "doubles" is , "triples" is , "halves" is . Then raise it to the number of periods, e.g. tripling for hours is .
Exponential change means multiplying by the same factor every period. After periods: (No knowledge of is required — this is just repeated proportional change.)
What multiplier represents a 4% increase per year?
A colony of bacteria starts with cells and triples every hour. How many cells are there after hours?
