Algebra 2 Quiz: Recognize Percent Growth Or Decay
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Recognize Percent Growth Or DecayQuestion 1 of 20

A company tracks the number of active users each month:

Month: 0, 1, 2, 3 Users: 12,000; 13,200; 14,520; 15,972

Does this represent constant percent change, and if so, what is the percent growth rate per month?

No; it's linear because it increases by 1,200 each month
Yes; 12% growth per month (growth factor b=1.12b=1.12)
Yes; 10% growth per month (growth factor b=1.10b=1.10)
Yes; 1.1% growth per month (because ratios are 1.1)
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Algebra 2 Quiz

Algebra 2 Quiz: Recognize Percent Growth Or Decay

Practice Recognize Percent Growth Or Decay in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Recognize Percent Growth Or Decay, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A company tracks the number of active users each month:

Month: 0, 1, 2, 3 Users: 12,000; 13,200; 14,520; 15,972

Does this represent constant percent change, and if so, what is the percent growth rate per month?

  1. No; it's linear because it increases by 1,200 each month
  2. Yes; 12% growth per month (growth factor b=1.12b=1.12)
  3. Yes; 10% growth per month (growth factor b=1.10b=1.10) (correct answer)
  4. Yes; 1.1% growth per month (because ratios are 1.1)
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! Applying the ratio test: 13,200/12,000 = 1.1, 14,520/13,200 = 1.1, 15,972/14,520 = 1.1, so constant ratios at 1.1 confirm exponential growth with b=1.10, and percent rate (1.10 - 1)*100% = 10% per month. Choice C correctly identifies yes, 10% growth per month through these constant ratios and rate calculation. A distractor like choice A might confuse it with linear due to increasing values, but the differences (1,200; 1,320; 1,452) aren't constant, so it's not linear—nice work distinguishing them! The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 2

A laptop loses value each year due to depreciation. Its value is $1200 after 0 years, $1020 after 1 year, $867 after 2 years, and $736.95 after 3 years.

From the data, what is the percent rate of change per year, and is it growth or decay?

  1. 10% growth per year
  2. 15% decay per year (correct answer)
  3. 15% growth per year
  4. 0.85% decay per year
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! Let's calculate the ratios: 1020/1200 = 0.85, 867/1020 = 0.85, 736.95/867 = 0.85. All ratios equal 0.85, confirming exponential decay with base 0.85. Choice B correctly identifies 15% decay per year through constant ratios: base 0.85 means each year retains 85% of previous value, losing 15% (100% - 85% = 15%). Choice C incorrectly claims 15% growth—but ratios less than 1 indicate decay, not growth! The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 3

A recycling bin contains 500 pounds of material. Each day, 6% of the material is removed. Which statement correctly identifies the constant percent rate and the exponential model for the amount remaining after dd days?

  1. 6% decay; M(d)=500(0.94)dM(d)=500(0.94)^d (correct answer)
  2. 94% decay; M(d)=500(0.06)dM(d)=500(0.06)^d
  3. 6% growth; M(d)=500(1.06)dM(d)=500(1.06)^d
  4. Linear decay; M(d)=500−0.06dM(d)=500-0.06d
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! If 6% is removed each day, then 94% remains (100% - 6% = 94%). This means we multiply by 0.94 each day: Day 0: 500 pounds, Day 1: 500 × 0.94, Day 2: 500 × (0.94)², giving us M(d) = 500(0.94)^d. Choice B correctly identifies 6% decay with the model M(d) = 500(0.94)^d (base 0.94 represents 94% remaining). Choice A incorrectly suggests growth when material is being removed; Choice C misinterprets the situation as 94% decay (which would leave only 6%); Choice D suggests a linear model which doesn't match percent removal. The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 4

Two savings plans are described below.

Plan 1: Start with $2000 and add $50 each month. Plan 2: Start with $2000 and increase the balance by 2% each month.

Which plan shows constant percent change (exponential), and which shows constant additive change (linear)?

  1. Plan 1 is exponential; Plan 2 is linear
  2. Both plans are exponential because both increase each month
  3. Both plans are linear because they both change monthly
  4. Plan 2 is exponential; Plan 1 is linear (correct answer)
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! Plan 1 adds $50 each month—this is constant additive change, making it linear. Plan 2 increases the balance by 2% each month, meaning each month's balance is 102% of the previous (multiply by 1.02)—this is constant percent change, making it exponential. Choice D correctly identifies Plan 2 as exponential (constant percent change) and Plan 1 as linear (constant additive change). Choice A reverses these classifications, missing that "add $50" signals linear while "increase by 2%" signals exponential. The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms!

Question 5

Two phone plans track the number of users over months. Plan A adds 250 users each month. Plan B increases the number of users by 5% each month. Which statement is correct about the type of change for each plan?

  1. Plan A is exponential; Plan B is linear
  2. Both plans are linear because they change each month
  3. Plan A is linear (constant additive); Plan B is exponential (constant percent) (correct answer)
  4. Both plans are exponential because they both increase
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! Plan A adds 250 users each month—this is a constant amount added, making it linear (if starting at N₀, then N(t) = N₀ + 250t). Plan B increases by 5% each month—this is a constant percent change, making it exponential (if starting at N₀, then N(t) = N₀(1.05)^t). Choice C correctly identifies Plan A as linear (constant additive change of +250) and Plan B as exponential (constant percent change of 5%). Choice A reverses the classifications; Choice B incorrectly claims both are linear; Choice D incorrectly claims both are exponential just because they increase. The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 6

A scientist records a sample's mass (in grams) after each processing step:

Step nn: 0, 1, 2, 3 Mass M(n)M(n): 200, 180, 162, 145.8

Classify the pattern and identify the constant percent rate per step.

  1. Neither; it is linear because it decreases by 20 grams each step.
  2. Exponential growth at 10% per step.
  3. Exponential decay at 0.90% per step.
  4. Exponential decay at 10% per step. (correct answer)
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! To check, compute the ratios from the table: 180/200 = 0.9, 162/180 = 0.9, and 145.8/162 = 0.9, showing a constant ratio of 0.9, fitting exponential decay. Choice C correctly identifies exponential decay at 10% per step through constant ratios of 0.9 (since 1 - 0.9 = 0.1 or 10%). One distractor suggests linear by 20 grams, but differences are 20, 18, 16.2—not constant; excellent work spotting the ratio pattern instead! The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 7

A city's population is recorded every 5 years:

Year: 0, 5, 10, 15 Population: 50,000; 53,000; 56,000; 59,000

Classify the pattern as exponential growth, exponential decay, or neither. (Hint: compare differences vs ratios.)

  1. Exponential growth, because the ratio is constant.
  2. Exponential decay, because the population increases by smaller amounts each interval.
  3. Neither; it shows approximately constant additive change (more linear than exponential). (correct answer)
  4. Exponential growth at 6% per 5 years.
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! Differences are +3000 each (constant, linear); ratios are 53000/50000=1.06, 56000/53000≈1.0566, 59000/56000≈1.0536—not constant, so not exponential. Choice C correctly identifies neither (more linear than exponential) through comparing constant differences vs. non-constant ratios. One distractor suggests exponential growth because it increases, but ratios aren't constant—nice try, but always verify both tests! The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 8

A radioactive sample's mass is recorded every day:

Day: 0, 1, 2, 3 Mass (g): 50, 46, 42, 38

Does this represent constant percent decay? (Hint: constant percent change means a constant ratio, not a constant difference.)

  1. Yes; it is exponential decay because it decreases by 4 g each day
  2. Yes; 4% decay per day because it decreases by 4 each day
  3. Yes; 8% decay per day because 46/50=0.9246/50=0.92
  4. No; the ratios 46/5046/50, 42/4642/46, and 38/4238/42 are not all equal (correct answer)
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! Let's check the ratios: 46/50 = 0.92, 42/46 ≈ 0.913, 38/42 ≈ 0.905. The ratios are not constant (0.92 ≠ 0.913 ≠ 0.905), so this is NOT exponential decay. Instead, the differences are constant: 50-46=4, 46-42=4, 42-38=4, indicating linear decay. Choice B correctly identifies that the ratios are not all equal, confirming this is not constant percent decay. Choice C incorrectly claims exponential decay based on the constant difference of 4g, which actually indicates linear change. The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms!

Question 9

An investment account follows the model A(t)=2500(1.06)tA(t)=2500(1.06)^t, where tt is in years.

Identify the constant percent rate and classify the change.

  1. Exponential decay at 6% per year
  2. Linear growth of 0.06⋅2500=1500.06\cdot 2500=150 dollars per year
  3. Exponential growth at 6% per year (correct answer)
  4. Exponential growth at 1.06% per year
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! From the formula A(t) = 2500(1.06)^t, we identify the base as 1.06. Since 1.06 > 1, this represents exponential growth. The percent rate is calculated as: rate = base - 1 = 1.06 - 1 = 0.06 = 6%. Choice C correctly identifies exponential growth at 6% per year through base analysis: base 1.06 means multiplying by 1.06 each year, which is 106% of previous value, representing 6% growth. Choice D incorrectly states 1.06% growth—confusing the base 1.06 with the percent rate; the base 1.06 corresponds to 6% growth, not 1.06% growth! The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 10

A gym membership fee changes over time according to the table.

Month mm: 0, 1, 2, 3 Fee F(m)F(m): 40, 44, 48, 52

Classify the change as exponential growth, exponential decay, or neither.​​

  1. Exponential growth because the fee increases each month
  2. Neither; it shows constant additive change (linear) (correct answer)
  3. Exponential decay because the ratios are less than 1
  4. Exponential growth at 10% per month
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! Let's check the differences: 44-40 = 4, 48-44 = 4, 52-48 = 4. The differences are constant at +4 per month, indicating linear change. Let's verify with ratios: 44/40 = 1.1, 48/44 ≈ 1.091, 52/48 ≈ 1.083. The ratios are not constant, confirming this is not exponential. Choice B correctly identifies this as neither exponential growth nor decay, but rather constant additive (linear) change. Choice A incorrectly claims exponential growth just because values increase, while choice D wrongly calculates a 10% growth rate when the ratios aren't even constant. The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms!

Question 11

A savings account is modeled by the function A(t)=2500(1.04)tA(t)=2500(1.04)^t, where tt is in years.

What is the constant percent rate of change, and is it growth or decay?

  1. 1.04% growth per year.
  2. It is linear because 25002500 is the starting amount.
  3. 4% decay per year.
  4. 4% growth per year. (correct answer)
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! The function A(t) = 2500(1.04)^t is already in exponential form, with base b=1.04, indicating a constant multiplicative factor of 1.04 per year. Choice A correctly identifies 4% growth per year through base analysis (since 1.04 - 1 = 0.04 or 4%). One distractor confuses the rate with 1.04%, but the percent rate is (b - 1) * 100%, so 0.04 is 4%, not 1.04%—keep practicing to spot this difference! The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 12

A radioactive sample has mass values recorded each day.

Day dd: 0, 1, 2, 3 Mass m(d)m(d) (g): 200, 170, 144.5, 122.825

From the table, determine whether the sample shows constant percent change. If it does, find the percent decay rate per day.​​

  1. Yes; 15% decay per day (correct answer)
  2. Yes; 17% decay per day
  3. Yes; 15% growth per day
  4. No; it is linear because it decreases by 30 g per day
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! Let's calculate the ratios: 170/200 = 0.85, 144.5/170 = 0.85, 122.825/144.5 = 0.85. All ratios equal 0.85, confirming exponential decay with base 0.85. Since 0.85 = 1 - 0.15, the decay rate is 15% per day (the sample retains 85% of its mass each day). Choice A correctly identifies 15% decay per day. Choice D incorrectly claims linear decay—the differences are 30, 25.5, 21.675, which aren't constant, while the ratios are constant at 0.85, confirming exponential decay. The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms!

Question 13

Which situation shows constant percent change (exponential) rather than constant additive change (linear)?​

  1. A population increases by 4% each year (correct answer)
  2. A tank is filled by adding 3 liters every minute
  3. A salary increases by $50 each month
  4. A runner improves time by 2 seconds each week
Explanation: This question tests your ability to recognize exponential growth or decay—situations where a quantity changes by a constant percent (not constant amount) per time interval. Constant percent growth means multiplying by the same factor greater than 1 each interval: if something grows 5% per year, each year's value is 105% of the previous (multiply by 1.05). Constant percent decay means multiplying by a factor between 0 and 1: if something decreases 15% per year, each year retains 85% of previous (multiply by 0.85). The key test: calculate ratios of consecutive values. If ratios are constant, it's exponential with that ratio as the growth/decay factor! In the situations, choice C describes a 4% increase each year, which means multiplying by 1.04 annually, a constant percent change. Choice C correctly identifies the exponential growth through the explicit percent language, distinguishing it from additive changes in the others. Distractors like choices A, B, and D use fixed amounts (3 liters, $50, 2 seconds), which are linear—excellent observation on the wording difference! The ratio test for exponential: (1) from table, divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, (2) if all equal → exponential with that ratio as base b, (3) if b > 1 → growth; if 0 < b < 1 → decay, (4) calculate percent rate: r = b - 1, convert to percent. Example: ratios all 1.06 → base 1.06 → growth → rate = 1.06 - 1 = 0.06 = 6% per interval. This systematic check identifies exponential patterns reliably! Don't confuse exponential with linear: linear adds the same amount each time (constant differences like +50, +50, +50), exponential multiplies by same factor (constant ratios like ×1.1, ×1.1, ×1.1). Both are patterns of regular change, but different mechanisms! Check both: if differences constant → linear. If ratios constant → exponential. Usually only one pattern holds. The language helps too: 'grows by $50 per year' = linear (additive), 'grows by 5% per year' = exponential (multiplicative)!

Question 14

A bank account earns interest such that the balance after tt years is given by B(t)=5000(1.045)tB(t) = 5000(1.045)^t. A savings bond increases in value according to V(t)=3000+180tV(t) = 3000 + 180t. Which statement best describes the growth patterns of these two investments?

  1. The bank account grows by a constant percent rate of 4.5% per year, while the bond grows by a constant dollar amount of $180 per year (correct answer)
  2. The bank account grows by a constant dollar amount of $225 per year, while the bond grows by a constant percent rate of 6% per year
  3. Both investments grow by constant percent rates, with the account at 4.5% and the bond at 18% per year respectively
  4. The bank account grows by a constant percent rate of 45% per year, while the bond grows by a constant percent rate of 180% per year
Explanation: The bank account function B(t)=5000(1.045)tB(t) = 5000(1.045)^t is exponential with base 1.045, indicating 4.5% growth per year. The bond function V(t)=3000+180tV(t) = 3000 + 180t is linear, showing constant dollar increases of $180 yearly. Choice B incorrectly treats the exponential as linear. Choice C misinterprets the linear function as exponential. Choice D confuses the growth factor with the growth rate.

Question 15

The value of a car depreciates according to V(t)=25000(0.82)tV(t) = 25000(0.82)^t, where tt is years since purchase. The owner also considers that inflation affects purchasing power at 3% annually. Relative to inflation, how does the car's value change each year?

  1. The car loses 18% of its value annually, but when adjusted for 3% inflation, the real depreciation rate becomes 21% per year
  2. The car depreciates at 18% yearly while inflation is 3% yearly, resulting in a net real depreciation of approximately 15% annually
  3. After accounting for inflation, the car's real value decreases by approximately 20.4% per year since 0.821.03≈0.796\frac{0.82}{1.03} \approx 0.796 (correct answer)
  4. The inflation-adjusted depreciation is 14.6% per year because the nominal rate of 18% minus inflation of 3% equals 15% net loss
Explanation: The car retains 82% of its value annually while inflation is 3%. The real value retention rate is 0.821.03≈0.796\frac{0.82}{1.03} \approx 0.796, meaning about 20.4% real depreciation yearly. Choice A incorrectly adds the rates. Choice B incorrectly subtracts rates linearly. Choice D also uses incorrect linear subtraction and provides the wrong final percentage.

Question 16

A researcher observes that a bacterial culture triples every 4 hours. If the initial population is 500 bacteria, and she wants to model this with the form N(t)=500⋅btN(t) = 500 \cdot b^t where tt is in hours, what is the hourly percent growth rate of this culture?

  1. The culture grows at a rate of 75% per hour since it triples every 4 hours and 3÷4=0.753 \div 4 = 0.75
  2. The hourly growth rate is approximately 31.6% since b=31/4≈1.316b = 3^{1/4} \approx 1.316, giving 31.6% increase per hour (correct answer)
  3. The bacteria increase by exactly 300% per hour because tripling represents a 300% increase from the original amount
  4. The growth rate is 25% per hour since the population increases by one-fourth of the tripling rate each hour
Explanation: Since the culture triples every 4 hours, we have 3=b43 = b^4, so b=31/4≈1.316b = 3^{1/4} \approx 1.316. This means 31.6% growth per hour. Choice A incorrectly divides the multiple by time. Choice C confuses tripling (200% increase) with 300% and misapplies the timeframe. Choice D arbitrarily assigns 25% without proper calculation.

Question 17

A radioactive isotope decays such that 12.5% of the original amount remains after 30 days. A chemist needs to determine the daily decay rate to predict when only 1% will remain. What is the constant daily percent decay rate?

  1. The decay rate is 4.2% daily since the half-life appears to be about 10 days and 21/10≈1.0422^{1/10} \approx 1.042 giving 4.2% daily decay
  2. The daily decay rate is exactly 2.5% because 12.5% remaining means 87.5% decayed over 30 days, giving 87.5÷30=2.92%87.5 \div 30 = 2.92\% daily
  3. The isotope loses approximately 2.92% of its mass each day based on linear distribution of the total 87.5% loss over 30 days
  4. The isotope decays at approximately 6.9% per day since 0.125=(0.931)300.125 = (0.931)^{30} and 1−0.931=0.0691 - 0.931 = 0.069 (correct answer)
Explanation: When you encounter radioactive decay problems, remember that decay follows an exponential model, not a linear one. The amount remaining follows the formula A=A0(1−r)tA = A_0(1-r)^t, where rr is the daily decay rate. To find the daily decay rate, you need to work backwards from the given information. You know that 12.5% (or 0.125) remains after 30 days, so: 0.125=(1−r)300.125 = (1-r)^{30} Taking the 30th root of both sides: (1−r)=(0.125)1/30=0.931(1-r) = (0.125)^{1/30} = 0.931 Therefore: r=1−0.931=0.069r = 1 - 0.931 = 0.069 or about 6.9% daily decay. Answer A incorrectly assumes the half-life is 10 days and uses an inappropriate formula. The actual half-life would be much shorter given that only 12.5% remains after 30 days. Answer B makes the critical error of assuming linear decay, simply dividing the total percentage lost by the number of days. This ignores the exponential nature of radioactive decay. Answer C commits the same linear thinking mistake as B, treating decay as if the same absolute amount is lost each day rather than the same percentage of what remains. The key study tip for exponential decay problems: never use linear division to find rates. Always use the exponential formula (1−r)t=fraction remaining(1-r)^t = \text{fraction remaining} and solve for rr by taking the appropriate root. Radioactive decay is inherently exponential because each day you lose a percentage of what's left, not a fixed amount.

Question 18

A population study tracks two cities over time. City A has a population that can be modeled by PA(t)=50000⋅2t/10P_A(t) = 50000 \cdot 2^{t/10}, where tt is years since 2020. City B's population follows PB(t)=45000(1.071)tP_B(t) = 45000(1.071)^t. After analyzing both models, which conclusion is most accurate?

  1. City A doubles every 10 years with an annual growth rate of approximately 7.2%, while City B grows at exactly 7.1% per year (correct answer)
  2. City A grows at 20% per year, while City B grows at 7.1% per year, making City A's growth rate higher throughout the period
  3. Both cities have exponential growth, but City A's rate changes over time while City B maintains a constant 7.1% annual rate
  4. City A has a constant growth rate of 10% per year, while City B grows at 7.1% per year with compounding effects
Explanation: City A's model PA(t)=50000⋅2t/10P_A(t) = 50000 \cdot 2^{t/10} shows the population doubles every 10 years. Converting to annual rate: 21/10≈1.0722^{1/10} \approx 1.072, so about 7.2% annually. City B grows at exactly 7.1% per year from the factor 1.071. Choice B misinterprets the exponent structure. Choice C incorrectly suggests City A's rate varies. Choice D misreads the doubling period as a growth rate.

Question 19

An investment advisor presents two options: Investment X grows according to f(t)=8000(0.85)tf(t) = 8000(0.85)^t and Investment Y follows g(t)=12000(1.15)tg(t) = 12000(1.15)^t, where tt represents years. A client asks about the percent change behavior of these investments. What should the advisor conclude?

  1. Investment X decreases by 15% annually while Investment Y increases by 15% annually, representing symmetric but opposite growth patterns
  2. Investment X loses 85% of its value each year while Investment Y gains 115% of its value each year over time
  3. Investment X declines by exactly 15% per year while Investment Y appreciates by exactly 15% per year consistently (correct answer)
  4. Investment X has variable decay rates while Investment Y maintains steady growth of 1.15% per year throughout the period
Explanation: Investment X has factor 0.85, meaning it retains 85% each year (loses 15%). Investment Y has factor 1.15, meaning it grows to 115% each year (gains 15%). Both show constant percent rates. Choice A incorrectly states Y decreases. Choice B misinterprets the factors as percentages lost/gained rather than multipliers. Choice D incorrectly suggests variable rates and misreads 1.15 as 1.15%.

Question 20

A marketing team compares two advertising campaigns. Campaign A reaches RA(t)=1000⋅3t/2R_A(t) = 1000 \cdot 3^{t/2} people after tt days, while Campaign B reaches RB(t)=800(1.73)tR_B(t) = 800(1.73)^t people. Both campaigns aim to maximize daily percent growth in reach. Which analysis is correct?

  1. Campaign A grows at approximately 73% per day since 31/2≈1.733^{1/2} \approx 1.73, matching Campaign B's daily growth rate exactly
  2. Campaign A doubles its reach every 2 days for about 41.4% daily growth, while Campaign B grows at 73% daily, making B more aggressive (correct answer)
  3. Campaign A has a 50% daily growth rate because it involves t/2t/2, while Campaign B maintains exactly 73% daily growth throughout
  4. Campaign A triples every 2 days giving 150% daily growth, while Campaign B grows at 73% daily, so A is significantly more effective
Explanation: Campaign A triples every 2 days, so daily factor is 31/2≈1.4143^{1/2} \approx 1.414, giving about 41.4% daily growth. Campaign B grows at 73% daily from factor 1.73. Choice A correctly calculates A's rate but wrongly claims they match. Choice C misinterprets the exponent structure. Choice D confuses tripling (200% increase) with 150% and misapplies the timeframe.