SAT exponential growth and decay: the base is 1 + r, not r
The town grows 3% a year. Two of the four choices contain 0.03 and 1.03, and choosing between them is the entire question.
Why it happens
A percent change describes the change; the base of an exponential model describes the whole amount after the change. Writing 0.03 models a population that loses 97% a year. The two forms look so alike that the error survives right through to the answer.
See the trap
A town of 4,500 people grows 3% a year. Write the population after y years.
P = 4,500(0.03)^y — the percent inserted directly as the base.
P = 4,500(1.03)^y. Growth adds to 1; decay subtracts from it — a 3% decline would be 0.97, not −0.03.
How to actually fix it
- 1Write the base as 1 ± r before anything else: 1.03 for growth, 0.97 for decay.
- 2Test one period in your head. 4,500 × 1.03 = 4,635 is plausible; 4,500 × 0.03 = 135 is not.
- 3Check the exponent's units — if the rate is monthly and the time is in years, the exponent needs converting too.
Practice this trap · 9 real questions
Each of these is a question from the free bank that turns on the mistake above — the trap is the reason its wrong answer looks right. Easiest first.
Showing 8 of 9. Keep going in Advanced Math →
FAQ
What if the exponent is a fraction like t/330?
That is the model saying one full growth step takes 330 units of time. The base still tells you what happens over one step; the fraction tells you how long a step lasts — which is exactly what doubling-time questions are asking for.
How do I stop repeating this mistake?
Reading the explanation isn't enough — your brain needs to beat the trap on a fresh question. Practice the skill in the kaoshen.co question bank: miss one and it is logged with the exact error, then brought back to you three days later and again after fourteen.
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