Skip to main content
Sign in Create a free account
Varsity Tutors
Sign in

Recognize Percent Growth or Decay - Algebra 1

  1. A car's value is $18,000\$18{,}000 when purchased and then decreases by 20% each year. Which statement correctly describes the type of change?
    Answer: Exponential decay because the value is multiplied by 0.800.80 each year This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. The context describes 'decreases by 20% each year.' Key phrase: 'decreases by 20%' directly tells us this is exponential decay with a constant percent rate. Each year, the quantity is multiplied by 1 - 0.20 = 0.80, making this exponential rather than linear. In one year, you have 80% of what you started with (original 100% minus 20%). Choice A correctly identifies this as exponential decay because the value is multiplied by 0.80 each year, which represents keeping 80% (losing 20%) of the previous year's value. Choice B says linear decay when it's actually exponential: it mentions '20% of the original price each year,' but exponential decay means 20% of the CURRENT value each year, not the original. Year 1: lose 20% of $18,000 = $3,600. Year 2: lose 20% of $14,400 = $2,880 (not $3,600 again!). The loss amount changes because it's a percent of the changing value. Real-world clue: exponential growth/decay contexts involve processes where the amount of change depends on how much you currently have: 'Car depreciates by 20% per year' means a car worth $18,000 loses $3,600, but next year when it's worth $14,400, it loses only $2,880—the change is smaller when the base is smaller. That's exponential! Linear is when you subtract the same amount regardless of current value.
  2. A town's population is modeled by P(t)=12000(1.02)tP(t)=12000(1.02)^t, where tt is in years. Which description is correct?
    Answer: Exponential growth: increases 2% per year This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent growth means the quantity is multiplied by the same factor greater than 1 each time period: if a population grows by 2% per year, it's multiplied by 1.02 each year (since 102% = 1 + 0.02 = 1.02 means 'keep all of what you had plus gain 2% more'). This creates exponential growth where the amount added each period gets larger because you're taking a percent of an increasing base! Looking at the function P(t) = 12000(1.02)t(1.02)^t: the base 1.02 is greater than 1, indicating exponential growth. The percent rate is calculated from r = 1.02 - 1 = 0.02 = 2% growth. This means each time t increases by 1, P is multiplied by 1.02, which is a 2% increase. Choice C correctly identifies this as exponential growth: increases 2% per year because the base 1.02 means the population grows by 2% annually. Choice D has the percent rate wrong: the base 1.02 doesn't mean 102% growth. When b = 1.02, we subtract 1 to get the rate: 1.02 - 1 = 0.02 = 2% growth. The base includes the original 100% (the '1') plus the growth rate (the '0.02'), so it's 2%, not 102%! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.02 → r = 0.02 → 2% growth. The subtraction of 1 is the crucial step!
  3. A charity's monthly donations follow this pattern: Month 1: $1000, Month 2: $1100, Month 3: $1200, Month 4: $1300. Is this exponential growth, exponential decay, or neither?
    Answer: Neither (it shows linear growth: constant additive change) This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. The key difference: linear growth adds the same amount each time (constant rate: +100, +100, +100), while exponential growth multiplies by the same percent each time (constant ratio: ×1.1, ×1.1, ×1.1). To check which: if differences are constant, it's linear; if ratios are constant, it's exponential. Example: 1000, 1100, 1200, 1300 has constant differences (+100) = linear. But 1000, 1100, 1210, 1331 has constant ratios (×1.1) = exponential! Let's check if this is exponential by finding ratios: From Month 1 to 2: 1100/1000 = 1.1. From Month 2 to 3: 1200/1100 = 1.091. From Month 3 to 4: 1300/1200 = 1.083. The ratios differ (1.1, 1.091, 1.083), so this is NOT exponential growth/decay. Let's check differences: 1100-1000 = 100, 1200-1100 = 100, 1300-1200 = 100. Constant differences of +100 confirm linear growth! Choice C correctly identifies this as neither exponential growth nor decay because it shows linear growth with constant additive change of $100 per month. Choice A confuses linear with exponential: it sees the pattern of increasing values and assumes exponential, but we need to check how they're changing. The differences are constant (+100 each month), not the ratios. Constant addition = linear, constant multiplication = exponential! Exponential vs linear quick-check: calculate both differences AND ratios. If differences are constant (like +100, +100, +100), it's linear. If ratios are constant (like ×1.1, ×1.1, ×1.1), it's exponential. Can't be both! This two-part check prevents confusion between the types.
  4. A savings account balance follows the function B(t)=500(1.05)tB(t)=500(1.05)^t, where tt is the number of years. Does this represent constant percent growth or decay, and what is the percent rate per year?
    Answer: Exponential growth at 5% per year This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent growth means the quantity is multiplied by the same factor greater than 1 each time period: if a population grows by 5% per year, it's multiplied by 1.05 each year (since 105% = 1 + 0.05 = 1.05 means 'keep all of what you had plus gain 5% more'). This creates exponential growth where the amount added each period gets larger because you're taking a percent of an increasing base! Looking at the function B(t) = 500(1.05)t(1.05)^t: the base 1.05 is greater than 1, indicating exponential growth. The percent rate is calculated from r = 1.05 - 1 = 0.05 = 5% growth. This means each time t increases by 1, B is multiplied by 1.05, which is a 5% increase. Choice B correctly identifies this as exponential growth with 5% per year because the base 1.05 > 1 and 1.05 - 1 = 0.05 = 5%. Choice D has the percent rate wrong: the base 1.05 doesn't mean 105% growth. When b = 1.05, we subtract 1 to get the rate: 1.05 - 1 = 0.05 = 5% growth. The base includes the original 100% (the '1') plus the growth rate (the '0.05'), so it's 5%, not 105%! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.12 → r = 0.12 → 12% growth. For decay: b = 0.95 → r = -0.05 → 5% decay (we usually state as positive '5% decay' rather than 'negative 5%'). The subtraction of 1 is the crucial step!
  5. The value of a laptop over time is shown by V(t)=1200(0.90)tV(t)=1200(0.90)^t, where tt is the number of years since purchase. Identify the constant percent change per year.
    Answer: 10% decay per year This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. Looking at the function V(t) = 1200(0.90)t(0.90)^t: the base 0.90 is less than 1, indicating exponential decay. The percent rate is calculated from r = 1 - 0.90 = 0.10 = 10% decay. This means each time t increases by 1, V is multiplied by 0.90, which is a 10% decrease. Choice C correctly identifies this as 10% decay per year because the base 0.90 < 1 and we lose 1 - 0.90 = 0.10 = 10% each year. Choice B says 90% decay when it's actually 10% decay: looking at the base 0.90, this means we keep 90% each year, so we lose only 10%, not 90%. When the base is 0.90, that's the fraction we keep, not the fraction we lose! Real-world clue: exponential growth/decay contexts involve processes where the amount of change depends on how much you currently have: 'Value decreases by 10% per year' means a laptop worth $1200 loses $120, but one worth $600 loses only $60—the change is smaller when the base is smaller. That's exponential! Linear is when you subtract the same amount regardless of current value.
  6. Which situation involves a constant percent rate of change (exponential), not a constant additive change (linear)?
    Answer: A bank account increases by 5% each year. This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. The key difference: linear growth adds the same amount each time (constant rate: +15, +15, +15), while exponential growth multiplies by the same percent each time (constant ratio: ×1.05, ×1.05, ×1.05). To check which: if differences are constant, it's linear; if ratios are constant, it's exponential. Let's contrast: Options A, B, and D all describe adding a fixed amount each period—constant additive change. But option C describes 'increases by 5% each year' which involves multiplying by the same percent each time—constant multiplicative change. The exponential case has constant ratios (each year multiply by 1.05), while linear cases would have constant differences (add same dollar amount). This is exponential growth! Choice C correctly identifies a constant percent rate situation because 'increases by 5% each year' means the account balance is multiplied by 1.05 annually, creating exponential growth. Choice A sees 'charges $15 more each month' and this is linear: it adds the same amount each time, not multiplies by a percent. Constant addition = linear, constant multiplication = exponential! Real-world clue: exponential growth/decay contexts involve processes where the amount of change depends on how much you currently have: 'Bank account increases by 5% per year' means $1000 gains $50, but $10,000 gains $500—the change is bigger when the base is bigger. That's exponential! Linear is when you add the same amount regardless of current size.
  7. A company's number of users increases by 15% each month. Which function represents this situation if the company starts with 2,000 users at month t=0t=0?
    Answer: U(t)=2000(1.15)tU(t)=2000(1.15)^t This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent growth means the quantity is multiplied by the same factor greater than 1 each time period: if a population grows by 5% per year, it's multiplied by 1.05 each year (since 105% = 1 + 0.05 = 1.05 means 'keep all of what you had plus gain 5% more'). This creates exponential growth where the amount added each period gets larger because you're taking a percent of an increasing base! The context describes 'increases by 15% each month.' Key phrase: 'increases by 15%' directly tells us this is exponential growth with a constant percent rate. Each month, the quantity is multiplied by 1 + 0.15 = 1.15, making this exponential rather than linear. In one month, you have 115% of what you started with (original 100% plus 15%). Choice C correctly identifies this as U(t) = 2000(1.15)t(1.15)^t because increasing by 15% means multiplying by 1.15 each month, and we start with 2000 users at t = 0. Choice A has the wrong base: 0.85 would mean decay by 15% (keeping only 85%), not growth by 15%. When something increases by 15%, we multiply by 1.15, not 0.85! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway!
  8. A plant's height (in cm) is recorded weekly: Week tt: 0, 1, 2, 3 Height H(t)H(t): 10, 15, 20, 25 Is this exponential growth, exponential decay, or linear growth?
    Answer: Linear growth because the differences are constant This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. The key difference: linear growth adds the same amount each time (constant rate: +50, +50, +50), while exponential growth multiplies by the same percent each time (constant ratio: ×1.1, ×1.1, ×1.1). To check which: if differences are constant, it's linear; if ratios are constant, it's exponential. Example: 100, 150, 200, 250 has constant differences (+50) = linear. But 100, 110, 121, 133.1 has constant ratios (×1.1) = exponential! Let's contrast: This table shows constant differences of +5 each week, which is linear growth. But an exponential case would have constant ratios, like ×1.5 each week (e.g., 10, 15, 22.5, 33.75). The linear case has constant differences, confirming it's not exponential growth or decay. Choice C correctly identifies this as linear growth because the differences are constant at +5, not ratios. Choice A says exponential growth because it increases each week, but that's confusing the outcome (increase) with the mechanism: constant addition = linear, constant multiplication = exponential! Exponential vs linear quick-check: calculate both differences AND ratios. If differences are constant (like +5, +5, +5), it's linear. If ratios are constant (like ×1.1, ×1.1, ×1.1), it's exponential. Can't be both! This two-part check prevents confusion between the types. The ratio test for exponential from a table: (1) Divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, etc., (2) If all ratios are equal, it's exponential and that ratio is your growth/decay factor b, (3) If b > 1, it's growth; if 0 < b < 1, it's decay, (4) Calculate percent rate: r = b - 1, convert to percent. Example: ratios all equal 1.06 → exponential growth, 6% per period. Easy!
  9. The value of a laptop is modeled by V(t)=900(0.85)tV(t)=900(0.85)^t, where tt is in years. What is the percent rate of change per year?
    Answer: 15% decay per year This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. Looking at the function V(t) = 900(0.85)t(0.85)^t: the base 0.85 is less than 1, indicating exponential decay. The percent rate is calculated from r = 0.85 - 1 = -0.15 = -15% decay. This means each time t increases by 1, V is multiplied by 0.85, which is a 15% decrease. Choice A correctly identifies this as 15% decay per year because the base 0.85 means we keep 85% of the value each year, losing 15%. Choice B has the percent rate wrong: the base 0.85 doesn't mean 85% decay. When b = 0.85, we subtract 1 to get the rate: 0.85 - 1 = -0.15 = 15% decay. The base represents what fraction remains (85%), not what's lost! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 0.85 → r = -0.15 → 15% decay (we usually state as positive '15% decay' rather than 'negative 15%'). The subtraction of 1 is the crucial step!
  10. Consider the function y=300(0.90)ty = 300(0.90)^t, where tt is measured in months. What is the constant percent change per month, and is it growth or decay?
    Answer: 10% decay per month This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. Looking at the function y = 300(0.90)t(0.90)^t: the base 0.90 is less than 1, indicating exponential decay. The percent rate is calculated from r = 0.90 - 1 = -0.10, but we state it as 10% decay (since decay rate is 1 - 0.90 = 0.10 = 10%). This means each time t increases by 1, y is multiplied by 0.90, which is a 10% decrease. Choice C correctly identifies this as 10% decay per month because the base 0.90 implies keeping 90%, so losing 10%. Choice B has the percent rate wrong: the base 0.90 doesn't mean 90% decay. When b = 0.90, we subtract from 1 to get the decay rate: 1 - 0.90 = 0.10 = 10% decay. The base represents the portion kept (90%), so the decay is the remaining 10%! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.12 → r = 0.12 → 12% growth. For decay: b = 0.95 → r = -0.05 → 5% decay (we usually state as positive '5% decay' rather than 'negative 5%'). The subtraction of 1 is the crucial step! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway!
  11. A bacteria culture has a mass of 200 mg at time t=0t=0 hours and grows by a constant percent rate of 10% each hour. Which statement best describes this change (and contrasts it with linear change)?
    Answer: Exponential growth: it multiplies by 1.101.10 each hour. This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent growth means the quantity is multiplied by the same factor greater than 1 each time period: if a population grows by 10% per hour, it's multiplied by 1.10 each hour (since 110% = 1 + 0.10 = 1.10 means 'keep all of what you had plus gain 10% more'). This creates exponential growth where the amount added each period gets larger because you're taking a percent of an increasing base! The context describes 'grows by a constant percent rate of 10% each hour.' Key phrase: 'grows by 10%' directly tells us this is exponential growth with a constant percent rate. Each hour, the quantity is multiplied by 1 + 0.10 = 1.10, making this exponential rather than linear. In one hour, you have 110% of what you started with (original 100% plus 10%). Choice C correctly identifies this as exponential growth because it states the bacteria multiplies by 1.10 each hour, which is exactly what 10% growth means. Choice A confuses exponential with linear: it sees the pattern of increasing values and assumes linear addition, but 10% growth means multiplying by 1.10, not adding 10 mg. Constant addition = linear, constant multiplication = exponential! Real-world clue: exponential growth/decay contexts involve processes where the amount of change depends on how much you currently have: 'Bacteria grows by 10% per hour' means 200 mg gains 20 mg in the first hour, but 220 mg gains 22 mg in the second hour—the change is bigger when the base is bigger. That's exponential!
  12. A radioactive substance has a half-life of 6 hours. Starting with 320 grams, how much will remain after 18 hours, and what percent of the original amount is this?
    Answer: 40 grams, which is 12.5% of the original amount When you encounter radioactive decay problems, you're dealing with exponential decay where the amount halves at regular intervals. The key is understanding what "half-life" means and systematically working through each time period. Half-life means that every 6 hours, exactly half of the substance remains. Starting with 320 grams, let's trace what happens: - After 6 hours (1 half-life): 320÷2=160320 ÷ 2 = 160 grams - After 12 hours (2 half-lives): 160÷2=80160 ÷ 2 = 80 grams - After 18 hours (3 half-lives): 80÷2=4080 ÷ 2 = 40 grams To find the percentage: 40320×100%=12.5%\frac{40}{320} × 100\% = 12.5\% Choice A (80 grams, 25%) represents what remains after only 2 half-lives (12 hours), not 3 half-lives. This is a common error when students miscount the time periods. Choice C (53.33 grams, 16.67%) suggests using incorrect division or attempting to use a continuous decay formula inappropriately for discrete half-life periods. Choice D (160 grams, 50%) shows the amount after just 1 half-life (6 hours), indicating the student stopped calculating too early. The correct answer is B: 40 grams, which is 12.5% of the original amount. Study tip: For half-life problems, always count how many complete half-life periods fit into the total time, then repeatedly divide by 2 that many times. Drawing a timeline can help you avoid miscounting periods.
  13. A medication amount in the bloodstream is measured every hour: tt (hours): 0, 1, 2, 3 A(t)A(t) (mg): 80, 68, 57.8, 49.13 Is this exponential decay, exponential growth, or neither?
    Answer: Exponential decay This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. From a table, to identify exponential with constant percent rate: divide consecutive y-values to find ratios. If y₂/y₁ = y₃/y₂ = y₄/y₃ = same number, that's your growth/decay factor b. If b > 1 (like 1.08), it's growth at (b-1)×100% = 8%. If 0 < b < 1 (like 0.92), it's decay at (1-b)×100% = 8% decay. Let's check if this is exponential by finding ratios: From hour 0 to 1: 68/80 = 0.85. From hour 1 to 2: 57.8/68 = 0.85. From hour 2 to 3: 49.13/57.8 = 0.85. All ratios equal 0.85, confirming exponential! Since 0.85 is less than 1, this is decay. The percent rate is 1 - 0.85 = 0.15 = 15% decay. Choice C correctly identifies this as exponential decay because all consecutive ratios equal 0.85, which is less than 1, indicating the medication amount is multiplied by 0.85 (loses 15%) each hour. Choice B says linear decay when it's actually exponential decay: looking at the pattern, the differences are NOT constant (80 - 68 = 12; 68 - 57.8 = 10.2; 57.8 - 49.13 = 8.67), but the ratios ARE constant (all 0.85). When the base is between 0 and 1, that's exponential decay! Exponential vs linear quick-check: calculate both differences AND ratios. If differences are constant (like +5, +5, +5), it's linear. If ratios are constant (like ×1.1, ×1.1, ×1.1), it's exponential. Can't be both! This two-part check prevents confusion between the types.
  14. A wildlife population is recorded as follows: Year tt: 0, 1, 2, 3 Population: 500, 450, 405, 364.5 From the table, what is the percent change per year, and is it growth or decay?
    Answer: 10% decay per year This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. From a table, to identify exponential with constant percent rate: divide consecutive y-values to find ratios. If y₂/y₁ = y₃/y₂ = y₄/y₃ = same number, that's your growth/decay factor b. If b > 1 (like 1.08), it's growth at (b-1)×100% = 8%. If 0 < b < 1 (like 0.90), it's decay at (1-b)×100% = 10% decay. Let's check if this is exponential by finding ratios: From Year 0 to 1: 450/500 = 0.90. From Year 1 to 2: 405/450 = 0.90. From Year 2 to 3: 364.5/405 = 0.90. All ratios equal 0.90, confirming exponential! Since 0.90 is less than 1, this is decay. The percent rate is 1 - 0.90 = 0.10 = 10% decay. Choice A correctly identifies this as 10% decay per year because all consecutive ratios equal 0.90, showing the population retains 90% each year (loses 10%). Choice B says growth when it's actually decay: looking at the base 0.90, since 0.90 < 1, this is decay, not growth. When the base is between 0 and 1, that's exponential decay! The ratio test for exponential from a table: (1) Divide consecutive y-values: y₂/y₁, y₃/y₂, y₄/y₃, etc., (2) If all ratios are equal, it's exponential and that ratio is your growth/decay factor b, (3) If b > 1, it's growth; if 0 < b < 1, it's decay, (4) Calculate percent rate: for decay, r = 1 - b, convert to percent. Example: ratios all equal 0.90 → exponential decay, 10% per period. Easy!
  15. Which situation involves a constant percent rate of change (exponential), not a constant additive change (linear)?
    Answer: A savings account earns 5% interest each year This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. The key difference: linear growth adds the same amount each time (constant rate: +50, +50, +50), while exponential growth multiplies by the same percent each time (constant ratio: ×1.1, ×1.1, ×1.1). To check which: if differences are constant, it's linear; if ratios are constant, it's exponential. Example: 100, 150, 200, 250 has constant differences (+50) = linear. But 100, 110, 121, 133.1 has constant ratios (×1.1) = exponential! Let's contrast: Options B, C, D involve adding the same amount each time—constant additive change (e.g., +2 miles, +$3, +10 liters). But option A involves multiplying by the same percent each time—constant multiplicative change (5% interest means ×1.05 yearly). The exponential case has constant ratios, while a linear case would have constant differences. This is exponential growth! Choice A correctly identifies this as a savings account earning 5% interest each year because the context percent language shows constant percent change. Choice B confuses exponential with linear: it sees the pattern of increasing values and assumes exponential, but 'increases by 2 miles each week' means constant addition (+2, +2), not constant multiplication. Constant addition = linear, constant multiplication = exponential! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway! Real-world clue: exponential growth/decay contexts involve processes where the amount of change depends on how much you currently have: 'Population grows by 5% per year' means a population of 1000 gains 50, but a population of 10,000 gains 500—the change is bigger when the base is bigger. That's exponential! Linear is when you add the same amount regardless of current size.
  16. A population of bacteria doubles every 3 hours. If the initial population is 500 bacteria, what is the percent increase in the population after 9 hours compared to the initial population?
    Answer: 700% After 9 hours (3 doubling periods), the population grows by a factor of 23=82^3 = 8. The final population is 500×8=4000500 \times 8 = 4000. The percent increase is 4000−500500×100%=3500500×100%=700%\frac{4000 - 500}{500} \times 100\% = \frac{3500}{500} \times 100\% = 700\%. Choice A incorrectly uses only one doubling period. Choice B finds the final population as a percent of the initial (800%) but subtracts 100% incorrectly. Choice D gives the final population as a percent of initial without subtracting the original 100%.
  17. Which situation involves a constant percent rate of change (exponential), not a constant additive change (linear)? (a) A gym membership costs $40 per month plus a one-time $20 sign-up fee. (b) A car's value decreases by 10% each year. (c) A water tank is filled at 3 gallons per minute. (d) A plant grows 2 cm each week.​​
    Answer: Only (b) This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. The key difference: linear growth adds the same amount each time (constant rate: +50, +50, +50), while exponential growth multiplies by the same percent each time (constant ratio: ×1.1, ×1.1, ×1.1). To check which: if differences are constant, it's linear; if ratios are constant, it's exponential. Example: 100, 150, 200, 250 has constant differences (+50) = linear. But 100, 110, 121, 133.1 has constant ratios (×1.1) = exponential! Let's contrast: (a) gym membership costs $40 per month involves adding the same amount each time—constant additive change. (b) car's value decreases by 10% each year involves multiplying by the same percent each time—constant multiplicative change. (c) water tank filled at 3 gallons per minute is adding 3 gallons each time. (d) plant grows 2 cm each week is adding 2 cm each time. Only (b) has constant ratios, while the others have constant differences. This is exponential decay! Choice A correctly identifies only (b) as involving constant percent rate because 'decreases by 10% each year' means the car retains 90% of its value each year—multiply by 0.90, which is exponential decay. Choice C sees 'percent' in option (b) and correctly identifies it as exponential, but mistakenly includes (d) which says 'grows 2 cm each week'—that's adding the same length each time, not multiplying by the same percent. Growing by a fixed amount is linear, not exponential! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway!
  18. A medication amount in the bloodstream is modeled by A(t)=80(0.90)tA(t)=80(0.90)^t, where tt is measured in hours. What is the percent rate of change per hour?
    Answer: 10% decay per hour This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. Looking at the function A(t)=80(0.90)t(0.90)^t: the base 0.90 is less than 1, indicating exponential decay. The percent rate is calculated from r = 0.90 - 1 = -0.10 = 10% decay. This means each time t increases by 1, A is multiplied by 0.90, which is a 10% decrease. Choice C correctly identifies this as 10% decay per hour because the base < 1 and |r| = 0.10 confirms the rate. Choice B has the percent rate wrong: the base 0.90 doesn't mean 90% decay. When b = 0.90, we subtract 1 to get the rate: 0.90 - 1 = -0.10 = 10% decay. The base includes the remaining 90% (the '0.90'), so the decay is 10%, not 90%! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.12 → r = 0.12 → 12% growth. For decay: b = 0.95 → r = -0.05 → 5% decay (we usually state as positive '5% decay' rather than 'negative 5%'). The subtraction of 1 is the crucial step!
  19. A bacteria culture starts with 1000 cells and increases by 20% each hour. Which statement correctly identifies the growth factor bb and the percent rate rr per hour in the exponential form N(t)=a⋅btN(t)=a\cdot b^t?
    Answer: b=1.20b=1.20 and r=0.20r=0.20 (20% growth) This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent growth means the quantity is multiplied by the same factor greater than 1 each time period: if a population grows by 5% per year, it's multiplied by 1.05 each year (since 105% = 1 + 0.05 = 1.05 means 'keep all of what you had plus gain 5% more'). This creates exponential growth where the amount added each period gets larger because you're taking a percent of an increasing base! The context describes 'a bacteria culture starts with 1000 cells and increases by 20% each hour.' Key phrase: 'increases by 20% each hour' directly tells us this is exponential growth with a constant percent rate. Each hour, the quantity is multiplied by 1 + 0.20 = 1.20, making this exponential rather than linear. In one hour, you have 120% of what you started with (original 100% plus 20%). Choice B correctly identifies this as b=1.20 and r=0.20 (20% growth) because the growth factor includes the original 100% plus the 20% increase. Choice A says growth when it's actually decay (or vice versa): looking at the context, since it describes increase, this is growth, not decay. When the base is greater than 1, or when the context says 'increases,' that's exponential growth! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.12 → r = 0.12 → 12% growth. For decay: b = 0.95 → r = -0.05 → 5% decay (we usually state as positive '5% decay' rather than 'negative 5%'). The subtraction of 1 is the crucial step! Context language decoder for exponential: 'grows by X% per year,' 'decreases by X% per month,' 'X% interest compounded,' 'doubles every,' 'halves every,' 'increases X-fold' → all signal constant percent change (exponential). But 'adds $X per period' or 'increases by X units' → constant additive change (linear). The 'percent per period' pattern is the key giveaway!
  20. A population is modeled by P(t)=12,000⋅(0.98)tP(t)=12{,}000\cdot(0.98)^t, where tt is in years. Which statement is correct?
    Answer: Exponential decay at 2% per year This question tests your ability to recognize exponential relationships—situations where a quantity grows or decays by a constant percent rate per time period, which is very different from linear growth where you add the same amount each time. Constant percent decay means the quantity is multiplied by the same factor between 0 and 1 each time period: if a car depreciates by 15% per year, it's multiplied by 0.85 each year (since keeping 85% = 1 - 0.15 = 0.85 means 'you lose 15%'). The quantity shrinks exponentially, approaching but never quite reaching zero. Looking at the function P(t)=12,000·(0.98)t(0.98)^t: the base 0.98 is less than 1, indicating exponential decay. The percent rate is calculated from r = 0.98 - 1 = -0.02 = 2% decay. This means each time t increases by 1, P is multiplied by 0.98, which is a 2% decrease. Choice B correctly identifies this as exponential decay at 2% per year because the base <1 and |r|=0.02 confirms the rate. Choice D has the percent rate wrong: the base 0.98 doesn't mean 98% decay. When b = 0.98, we subtract 1 to get the rate: 0.98 - 1 = -0.02 = 2% decay. The base includes the remaining 98% (the '0.98'), so the decay is 2%, not 98%! To find percent rate from a growth/decay factor: (1) Identify b (the base or factor), (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: b = 1.12 → r = 0.12 → 12% growth. For decay: b = 0.95 → r = -0.05 → 5% decay (we usually state as positive '5% decay' rather than 'negative 5%'). The subtraction of 1 is the crucial step!

Start this quiz