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Algebra Help: Recognize Percent Growth Or Decay

Review real example questions for Recognize Percent Growth Or Decay in Algebra.

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A town's population is recorded each year:

Year $t$ : 0, 1, 2, 3 Population $P(t)$ : 20,000; 21,600; 23,328; 25,194.24

From the data, determine if there is constant percent change. If so, what is the percent rate per year?

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Question 1

A town's population is recorded each year:

Year $t$ : 0, 1, 2, 3 Population $P(t)$ : 20,000; 21,600; 23,328; 25,194.24

From the data, determine if there is constant percent change. If so, what is the percent rate per year?

Yes; exponential growth at 0.08% per year
Yes; exponential decay at 8% per year
No; it is linear because the population increases by 1,600 each year
Yes; exponential growth at 8% per year (correct answer)

Explanation: 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 year 0 to 1: 21,600/20,000 = 1.08. From year 1 to 2: 23,328/21,600 = 1.08. From year 2 to 3: 25,194.24/23,328 = 1.08. All ratios equal 1.08, confirming exponential! Since 1.08 is greater than 1, this is growth. The percent rate is 1.08 - 1 = 0.08 = 8%. Choice A correctly identifies this as exponential growth at 8% per year because all consecutive ratios equal 1.08 and 1.08 - 1 = 0.08 = 8% growth. Choice C confuses exponential with linear: it sees the pattern of increasing values and calculates the difference 21,600 - 20,000 = 1,600, but we need to check how they're changing. The differences are NOT constant (21,600 - 20,000 = 1,600; 23,328 - 21,600 = 1,728; 25,194.24 - 23,328 = 1,866.24), but the ratios ARE constant (all 1.08). Constant addition = linear, constant multiplication = exponential! 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!

Question 2

A bacteria culture starts with 1000 cells and increases by 20% each hour. Which statement correctly identifies the growth factor $b$ and the percent rate $r$ per hour in the exponential form $N(t)=a\cdot b^t$ ?

$b=1.02$ and $r=0.02$ (2% growth)
$b=0.80$ and $r=-0.20$ (20% decay)
$b=1.20$ and $r=0.20$ (20% growth) (correct answer)
$b=20$ and $r=20$ (20% growth)

Explanation: 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!

Question 3

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.

(d) A plant grows 2 cm each week.

Only (b) (correct answer)
Only (a) and (c)
Only (b) and (d)
All of them

Explanation: 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!

Question 4

A medication amount in the bloodstream is modeled by $A(t)=80(0.90)^t$ , where $t$ is measured in hours. What is the percent rate of change per hour?

0.90% decay per hour
10% decay per hour (correct answer)
10% growth per hour
90% decay per hour

Explanation: 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: 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!

Question 5

Consider the function $y = 300(0.90)^t$ , where $t$ is measured in months. What is the constant percent change per month, and is it growth or decay?

0.90% decay per month
90% decay per month
10% growth per month
10% decay per month (correct answer)

Explanation: 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: 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!

Question 6

A medication amount in the bloodstream decreases by 20% every hour. Which function represents the amount $A(t)$ after $t$ hours if $A(0)=80$ mg?

$A(t)=80(1.20)^t$
$A(t)=80(0.80)^t$ (correct answer)
$A(t)=80-20t$
$A(t)=80-0.20t$

Explanation: 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 medication decreases by 20% per hour, it's multiplied by 0.80 each hour (since keeping 80% = 1 - 0.20 = 0.80 means 'you lose 20%'). The quantity shrinks exponentially, approaching but never quite reaching zero. The context describes 'decreases by 20% every hour.' Key phrase: 'decreases by 20%' directly tells us this is exponential decay with a constant percent rate. Each hour, the quantity is multiplied by 1 - 0.20 = 0.80, making this exponential rather than linear. In one hour, you have 80% of what you started with (original 100% minus 20%). Choice B correctly identifies this as A(t) = 80(0.80)^t because decreasing by 20% means multiplying by 0.80 each hour, and the initial amount is 80 mg. Choice A says growth when it's actually decay: looking at the context, since it describes 'decreases,' this is decay, not growth. When the context says 'decreases,' 'depreciates,' or 'decays,' that's exponential decay with a base between 0 and 1! 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!

Question 7

A population is modeled by $P(t)=12{,}000\cdot(0.98)^t$ , where $t$ is in years. Which statement is correct?

Exponential growth at 2% per year
Neither; the base $0.98$ means 98% decay per year
Linear decay at 0.98 people per year
Exponential decay at 2% per year (correct answer)

Explanation: 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: 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!

Question 8

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 $t=0$ ?

$U(t)=2000(0.85)^t$
$U(t)=2000+0.15t$
$U(t)=2000(1.15)^t$ (correct answer)
$U(t)=2000(15)^t$

Explanation: 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 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!

Question 9

A car's value depreciates by 12% each year. After how many complete years will the car's value first drop below 50% of its original value?

4 years
5 years
6 years (correct answer)
7 years

Explanation: The car retains 88% = 0.88 of its value each year. We need $(0.88)^n < 0.5$ . Testing values: $(0.88)^4 ≈ 0.599$ , $(0.88)^5 ≈ 0.527$ , $(0.88)^6 ≈ 0.464$ . After 6 years, the value first drops below 50%. Choice A uses 4 years where value is still 59.9%. Choice B uses 5 years where value is still 52.7%. Choice D overshoots the requirement.

Question 10

A company's quarterly revenue data shows the following pattern: Q1: $200,000, Q2: $240,000, Q3: $288,000, Q4: $345,600.

Based on the revenue pattern shown, what type of growth model best describes this company's performance, and what would be the projected revenue for Q5?

Linear growth model with projected Q5 revenue of approximately $403,200
Exponential growth model with projected Q5 revenue of approximately $414,720 (correct answer)
Quadratic growth model with projected Q5 revenue of approximately $425,000
Exponential decay model with projected Q5 revenue of approximately $380,000

Explanation: Checking the ratios: $\frac{240,000}{200,000} = 1.2$ , $\frac{288,000}{240,000} = 1.2$ , $\frac{345,600}{288,000} = 1.2$ . The revenue grows by 20% each quarter, indicating exponential growth. Q5 projection: $345,600 \times 1.2 = 414,720$ . Choice A assumes linear growth (constant dollar increases). Choice C suggests quadratic growth (increasing rate of change). Choice D incorrectly identifies this as decay despite clear growth.