The key idea
In y=a(1+r)t, the quantity changes by the percent r each period. The exponent sets the period: bt/3 changes by the factor b every 3 units of t. To change units, rewrite the exponent: a yearly factor g is g1/12 per month.
The trap
Dividing a yearly percent by 12 to get a monthly percent, or reading 0.85 as an 85% decrease.
Can Desmos do it?
Desmos can evaluate a model or fit one to data. Interpreting the factor and converting periods is reasoning about the formula's structure.
College Board skill: Nonlinear functions · Advanced Math
One question from each level. The drill has 12 at every level, and moves you up after four right in a row.
Level 1 · Warm-up
A quantity is modeled by y=400(1.22)t, where t is the number of years since the first measurement. Which statement is true?
- AIt decreases by 22% each year.
- BIt increases by 22% each year.
- CIt increases by 88 each year.
- DIt increases by 122% each year.
Show the answer
Answer: BThe factor 1.22 is 1+0.22, so each year the quantity is multiplied by 1.22: a 22% increase. The change is a percent of the current amount, not a fixed amount.
Level 2 · Test level
The function V(t)=2000(0.78)t/3 gives a value t years after a purchase. Which statement is true?
- AThe value decreases by 22% every 3 years.
- BThe value decreases by 66% every 3 years.
- CThe value decreases by 22% every year.
- DThe value decreases by 7.33% every year.
Show the answer
Answer: AWhen t goes up by 3, the exponent 3t goes up by 1, so the value is multiplied by 0.78 once every 3 years: a 22% decrease every 3 years. It is not 7.33% per year, because percent changes compound rather than add.
Level 3 · Beyond the test
A quantity triples every 4 years. To the nearest whole percent, by what percent does it increase each year?
Show the answer
Answer: 32If the yearly factor is g, then 4 years of change give g4=3, so g=31/4≈1.3161. That is about a 32% increase each year. Dividing 200% by 4 would ignore compounding.
Practice all 36 questions
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