A reliable method
Translate a p percent increase to 1 + p/100 and a decrease to 1 − p/100. Match t to the stated time interval.
The common trap
A 20% increase followed by a 20% decrease does not restore the original amount: 1.2 × 0.8 = 0.96.
Make the connection
Two values at different times determine a factor only after you account for how many intervals separate them.
Compare equal increments and equal factors
A quantity starting at 50 and increasing by 10 each hour has values 50,60,70. A quantity starting at 50 and increasing by 20% each hour has values 50,60,72. The models agree for the first hour but diverge afterward. Compare differences and ratios across at least two intervals before choosing a model.
Respect the time interval
A population rises from 40 to 90 in two equal intervals under a constant positive factor r. Then 40r²=90, so r²=9/4 and r=3/2. The one-interval factor is 1.5; 2.25 is the factor for both intervals. The positive restriction selects the physically meaningful root.
Try, then check
An $800 item loses 25% of its value each year. After two years its modeled value is 800(3/4)²=$450. Subtracting $200 twice gives $400 and incorrectly fixes the loss to the original base. Explain why the second loss is $150 rather than $200.
A culture begins with 10 cells and doubles every hour. How many cells does the model predict after 3 hours?
- The model is N = 10 · 2^t.
- Three hours give three doubling intervals.
- N = 10 · 8 = 80.
Compare the other answer choices
Multiplied initial count by elapsed time rather than the exponential factor.
Multiplied the doubling factor by time instead of raising it to a power.
Counted four doubling intervals.
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.
Official mathematics skillsSiyavula Grade 10 mathematics