The decision and its conditions
Exponential change uses a fixed multiplicative factor for equal input increments. A retained fraction and a percent decrease describe the same change in different ways: retaining eighty percent means losing twenty percent, not eighty. To recover an initial amount after repeated halvings, undo each factor rather than adding a fixed quantity. Unequal-looking bases may be powers of a common base, allowing exponents to be compared. A table can reveal a multiplicative rate, but only the stated model lets you extend that pattern beyond observed entries.
Worked example
A model uses M(t)=500(4/5)^t for integer t≥0. What percent decrease occurs per unit increase in t? The retained fraction each step is 4/5. The lost fraction is 1−4/5=1/5. Convert that lost fraction to a percent: 20%.
A contrasting decision
An ideal sample halves once per hour. After 2 hours, 9 grams remain. How many grams were present initially? The retained fraction is (1/2)^2. Undo this by multiplying 9 by 2^2. The initial mass is 36 grams; reapplying 2 halvings returns 9.
Diagnose a competing method
The first example’s tempting answer 80 follows this mistake: Report the retained percentage as the decrease. The exponential factor is the retained share, not the decrease itself.
Transfer before revealing
Solve 4^x × 2 = 8^4 over the reals. Pause and identify the original restrictions. Solution: Rewrite both sides in base two. The exponent equation is 2x+1=12. Solving gives x=11/2.
Check the boundary of the method
Unequal-looking bases may be powers of a common base, allowing exponents to be compared. Undo repeated multiplication, not a fixed arithmetic decrease. Check the solution in the original statement. A transformed equation can generate candidates rather than equivalent solutions.
A model uses M(t)=500(4/5)^t for integer t≥0. What percent decrease occurs per unit increase in t?
- The retained fraction each step is 4/5.
- The lost fraction is 1−4/5=1/5.
- Convert that lost fraction to a percent: 20%.
Compare the other answer choices
Take the reciprocal of the decrease instead of converting units.
Report the retained percentage as the decrease.
Report the lost fraction without converting to percent.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
CLEP College Algebra exam overview