Recognize Percent Growth or Decay - Algebra 2

1. Which statement indicates constant percent growth relative to the current amount?

   "Increases by the same percent each interval". Same percent each interval indicates exponential growth.

2. What is the percent rate per interval if the common ratio in a table is $\frac{4}{5}$?

   $20\%$ decay per interval. Convert fraction to percent: $1 - \frac{4}{5} = 0.20 = 20\%$.

3. What condition on $b$ indicates exponential decay in $A(t)=A_0\cdot b^t$?

   $0<b<1$. Base between 0 and 1 means the quantity decreases.

4. What is the percent growth rate per year if a quantity "doubles every year"?

   $100\%$ growth per year. Doubling means $100\%$ increase from original amount.

5. What is the common ratio if values go $200, 150, 112.5$ at equal time steps?

   $0.75$. Divide consecutive terms: $\frac{150}{200} = \frac{112.5}{150} = 0.75$.

6. What type of change is indicated by "decreases by $12\%$ per year"?

   Constant percent decay (exponential decay). Percent change per unit time indicates exponential decay.

7. Identify the common ratio if a quantity grows by $30\%$ per interval.

   $1.3$. Add growth rate to 1: $1 + 0.30 = 1.3$.

8. Identify the base $b$ if "decays by $2\%$ each interval" is modeled by $A(t)=A_0\cdot b^t$.

   $b=0.98$. Subtract decay rate from 1: $1 - 0.02 = 0.98$.

9. Which phrase signals constant percent change: "decreases by $30$ each year" or "decreases by $30\%$ each year"?

   Decreases by $30\%$ each year. Percent change indicates exponential, not linear growth.

10. What is the multiplier for "grows by $3.5\%$ per day"?

    $1.035$. Add the percent rate as a decimal to 1.

11. Identify the model type for $P(t)=120\cdot(0.6)^t$.

    Exponential decay. Base $0.6 < 1$ indicates exponential decay.

12. Which phrase signals exponential decay: "retains $85\%$ each year" or "loses $85$ units each year"?

    Retains $85\%$ each year. Percentage retained indicates exponential decay model.

13. Identify whether $A(t+1)=1.04\,A(t)$ represents growth or decay.

    Growth. Multiplier $1.04 > 1$ indicates exponential growth.

14. What is the multiplier for "increases by $25\%$ per interval"?

    $1.25$. Add the percent rate as a decimal to 1.

15. Find the multiplier for "value halves every $6$ hours" per hour.

    $\left(\frac{1}{2}\right)^{\frac{1}{6}}$. Per-hour multiplier is the sixth root of $\frac{1}{2}$.

16. What is the per-interval percent change for $A(t)=A_0\cdot(0.72)^t$?

    $28\%$ decay per interval. Base $0.72$ means $28\%$ decay per interval.

17. What is the multiplier if a quantity "retains $85\%$ each year"?

    $0.85$. Retaining $85\%$ means multiplier is $0.85$.

18. What condition on $b$ indicates exponential growth in $A(t)=A_0\cdot b^t$?

    $b>1$. Base greater than 1 means the quantity increases.

19. What is the percent rate per step if values go $200,\ 150,\ 112.5$ at equal time steps?

    $25\%$ decay per step. Common ratio $0.75$ means $25\%$ decay per step.

20. Which situation is constant percent change: "value drops $200$ per year" or "value drops $20\%$ per year"?

    Value drops $20\%$ per year. Percent change indicates exponential, not linear decay.

21. Choose the model with constant percent change: $y=400\cdot(0.9)^t$ or $y=400-0.9t$.

    $y=400\cdot(0.9)^t$. Exponential form with base indicates percent change.

22. What is the percent rate per interval if the multiplier each interval is $1.08$?

    $8\%$ growth per interval. Subtract 1 from the multiplier and convert to percent.

23. Identify the base $b$ if "grows by $2\%$ each interval" is modeled by $A(t)=A_0\cdot b^t$.

    $b=1.02$. Add growth rate to 1: $1 + 0.02 = 1.02$.

24. What is the growth factor for a constant increase of $r\%$ per interval?

    $1+\frac{r}{100}$. Add the percent rate as a decimal to 1.

25. Which description matches $b=1$ in $A(t)=A_0\cdot b^t$?

    No change (constant value). Base equals 1 means no growth or decay occurs.

26. Identify the key recognition clue for constant additive change in a table of values.

    A constant difference between successive outputs. Constant difference indicates linear (additive) change.

27. Which statement indicates NOT constant percent change: "increases by $5$ each day" or "increases by $5\%$ each day"?

    Increases by $5$ each day. Fixed amount added indicates linear, not percent change.

28. Which is the correct base $b$ for "grows by $9\%$ per interval" in $A(t)=A_0\cdot b^t$?

    $b=1.09$. Add growth rate to 1: $1 + 0.09 = 1.09$.

29. What is the decay factor for a constant decrease of $r\%$ per interval?

    $1-\frac{r}{100}$. Subtract the percent rate as a decimal from 1.

30. Identify whether $A(t+1)=0.97\,A(t)$ represents growth or decay.

    Decay. Multiplier $0.97 < 1$ indicates exponential decay.

31. Which is the correct base $b$ for "decays by $9\%$ per interval" in $A(t)=A_0\cdot b^t$?

    $b=0.91$. Subtract decay rate from 1: $1 - 0.09 = 0.91$.

32. What is the multiplier if a quantity "is $120\%$ of the previous value each interval"?

    $1.2$. Being $120\%$ of previous means multiplier is $1.2$.

33. What is the percent rate per interval if $A(t+1)=0.995\,A(t)$?

    $0.5\%$ decay per interval. Subtract multiplier from 1 and convert to percent.

34. Which option indicates constant percent decay: $A(t)=A_0\cdot(0.85)^t$ or $A(t)=A_0-0.85t$?

    $A(t)=A_0\cdot(0.85)^t$. Base $0.85 < 1$ in exponential form indicates decay.

35. What is the multiplier for "decays by $18\%$ per hour"?

    $0.82$. Subtract the percent rate as a decimal from 1.

36. What is the multiplier for "decreases by $25\%$ per interval"?

    $0.75$. Subtract the percent rate as a decimal from 1.

37. Which option indicates exponential change: $A(t)=A_0+5t$ or $A(t)=A_0\cdot(1.05)^t$?

    $A(t)=A_0\cdot(1.05)^t$. Base raised to power $t$ indicates exponential change.

38. What phrase signals linear (not percent) change: "adds $15$ each week" or "multiplies by $1.15$ each week"?

    Adds $15$ each week (linear). Fixed amount added signals linear, not exponential change.

39. What is the percent rate per interval if the common ratio in a table is $\frac{5}{4}$?

    $25\%$ growth per interval. Convert fraction to percent: $\frac{5}{4} - 1 = 0.25 = 25\%$.

40. What is the per-interval percent change for $A(t)=A_0\cdot(1.15)^t$?

    $15\%$ growth per interval. Base $1.15$ means $15\%$ growth per interval.

41. What is the percent rate per interval if the multiplier each interval is $0.93$?

    $7\%$ decay per interval. Subtract the multiplier from 1 and convert to percent.

42. What is the percent rate per step if values go $80,\ 88,\ 96.8$ at equal time steps?

    $10\%$ growth per step. Common ratio $1.1$ means $10\%$ growth per step.

43. What is the percent rate per interval if $A(t+1)=1.005\,A(t)$?

    $0.5\%$ growth per interval. Subtract 1 from multiplier and convert to percent.

44. Which is constant percent change: ratios $\frac{A(t+1)}{A(t)}$ constant or differences $A(t+1)-A(t)$ constant?

    Ratios $\frac{A(t+1)}{A(t)}$ constant. Constant ratio between consecutive terms indicates percent change.

45. Identify the model type for $N(t)=50\cdot(1.2)^t$.

    Exponential growth. Base $1.2 > 1$ indicates exponential growth.

46. What is the common ratio if values go $200,\ 150,\ 112.5$ at equal time steps?

    $0.75$. Divide consecutive terms: $\frac{150}{200} = \frac{112.5}{150} = 0.75$.

47. Identify whether "doubles every year" represents constant percent growth per year.

    Yes; multiplier $2$ per year. Doubling means multiplying by $2$ each year.

48. Find the multiplier for "value is multiplied by $3$ every $2$ days" per day.

    $\sqrt{3}$. Per-day multiplier is the square root of $3$.

49. Identify the key recognition clue for constant percent change in a table of values.

    A constant ratio between successive outputs. Constant ratio indicates exponential (percent) change.

50. What type of change is indicated by "increases by $5\%$ each month"?

    Constant percent growth (exponential growth). Percent change per unit time indicates exponential growth.

51. What is the common ratio if values go $80,\ 88,\ 96.8$ at equal time steps?

    $1.1$. Divide consecutive terms: $\frac{88}{80} = \frac{96.8}{88} = 1.1$.

52. What is the exponential form for percent growth or decay over $t$ intervals?

    $A(t)=A_0\cdot b^t$. Standard exponential model with base $b$ and time $t$.

53. Identify the common ratio if a quantity decays by $30\%$ per interval.

    $0.7$. Subtract decay rate from 1: $1 - 0.30 = 0.7$.

54. Which situation is constant percent change: "$10\%$ interest yearly" or "$10$ dollars interest yearly"?

    $10\%$ interest yearly. Percent interest indicates exponential compound growth.

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