Doubling bacteria, decaying samples, growing investments — the test dresses one formula in a dozen costumes. Type the model once and every version falls.
The move: Write start · rate^time, then read.
Why it works: Growth and decay are repeated multiplication: start × (rate)^time. Graphed against a target value, “when does it reach…?” becomes a crossing you tap.
The pattern
- Type the model: start · (rate)^x
- Graph it or evaluate
- Tap the value or crossing you need
Worked example: A population of 500 grows 20% per year. What is the population after 3 years?
- Growth of 20% means multiply by 1.2 each year. Model: 500(1.2)ˣ.
f(x)=500(1.2)^x - Evaluate at year 3.
f(3) - 864. (Choice 800 is the “500 + 20%·3” linear trap — growth COMPOUNDS.)
Worked example 2: A sample of 80 grams has a half-life of 5 hours. How many grams remain after 15 hours?
- 15 hours = 3 half-lives. Model: 80 · (1/2)^(t/5).
f(t)=80 · ((1/2))^fract5 - Evaluate at t = 15: 80 → 40 → 20 → 10.
f(15)
Remember it on test day
Start × rate^time. That's the whole cue — say it when you see this question type and the steps follow.
When this isn't the move
Compound-interest questions with unusual compounding periods need the rate and exponent adjusted — set up the model carefully before typing.
Drill it until it's reflex
This move is one of Desvantage's 18 training modules: a plain-English lesson, a type-it-yourself walkthrough inside a live Desmos calculator, and a 100-question drill bank that deals ten fresh questions per round. The first two modules — plus the 12-question diagnostic — are free.