Algebra Quiz: Interpret Exponential Functions And Growth Rate

What this quiz covers

This quiz focuses on Interpret Exponential Functions And Growth Rate, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.

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.

Question 1

1. 0.03% growth
2. 103% growth
3. 1.03% growth
4. 3% growth (correct answer)

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. For the function P(t) = 800(1.03)^t, the base is 1.03. To find the percent rate, we calculate r = 1.03 - 1 = 0.03. Converting to percent: 0.03 × 100% = 3%. Since 1.03 is greater than 1, this is growth, specifically 3% growth per year. Choice B correctly identifies the percent rate as 3% growth by recognizing that the base 1.03 represents a 3% annual increase. Excellent! Choice C makes a common percent mistake: the base 1.03 doesn't mean 103% growth—it means 3% growth! The 1 represents 'what you already have' (100%), and the 0.03 is the additional 3%, for a total of 103% of the previous amount (which is 3% growth). To find the percent rate: (1) Identify the base b, (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: base is 1.03, so r = 1.03 - 1 = 0.03 = 3%. For decay like 0.97: r = 0.97 - 1 = -0.03 = -3%, which we call '3% decay.' Easy!

Question 2

1. $P(t)=1000(0.10)^t$
2. $P(t)=1000(1.10)^t$ (correct answer)
3. $P(t)=1100(1.10)^t$
4. $P(t)=1000(1.00)^t$

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form! The context tells us initial value is 1000 and rate is 10% increase. Converting the rate to decimal: 10% = 0.10. The growth factor is b = 1 + 0.10 = 1.10. So the exponential function is y = 1000·(1.10)^t. We can also write this as y = 1000(1 + 0.10)^t to show the rate explicitly! Choice B correctly identifies the function as y=1000(1.10)^t by showing correct reasoning. Excellent! Choice A confuses growth with decay (or vice versa): since the base 0.10 is less than 1, this is decay, not growth. An easy way to remember: bases bigger than 1 mean growing, bases between 0 and 1 mean shrinking! The form y = a(1 + r)^x makes the rate super obvious: if you see y = 500(1 + 0.08)^t, you can read the rate right off—it's 0.08 = 8%. But if you see y = 500(1.08)^t, you have to subtract 1 from the base: 1.08 - 1 = 0.08 = 8%. Same rate, just written differently!

Question 3

1. Neither (constant)
2. Exponential decay (correct answer)
3. Exponential growth
4. Linear decay

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.03), while the rate r is how much it's changing by percent (like 3%). They're related by b = 1 + r, so knowing one gives you the other! Looking at the function V(t) = 25000(0.88)^t, we check the base: 0.88 is less than 1, which means this is exponential decay. Think of it this way: each time t increases by 1, V is multiplied by 0.88, so V is getting smaller—that's decay! Choice C correctly identifies this as exponential decay because the base 0.88 < 1. Excellent! Choice A confuses growth with decay: since the base 0.88 is less than 1, this is decay, not growth. An easy way to remember: bases bigger than 1 mean growing, bases between 0 and 1 mean shrinking! Here's your growth/decay decision tree: (1) Look at the base b, (2) Is b > 1? That's growth. Is 0 < b < 1? That's decay. Is b = 1? No change. That's it! For example, 1.07 > 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 4

1. $y=100(1.08)^t$
2. $y=100(1.15)^t$
3. $y=100(0.90)^t$ (correct answer)
4. $y=100(1.00)^t$

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. Looking at each function, we check the bases: A has base 1.08 > 1 (growth), B has base 0.90 < 1 (decay), C has base 1.00 = 1 (constant), D has base 1.15 > 1 (growth). Only choice B has a base less than 1, making it exponential decay. Choice B correctly identifies y = 100(0.90)^t as exponential decay because the base 0.90 < 1. Excellent! Choices A and D have bases greater than 1, so they represent growth, not decay. Choice C has base exactly 1, which means no change—it stays constant at 100. Here's your growth/decay decision tree: (1) Look at the base b, (2) Is b > 1? That's growth. Is 0 < b < 1? That's decay. Is b = 1? No change. That's it! For example, 1.07 > 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 5

1. 20% growth
2. 80% decay
3. 0.20% decay
4. 20% decay (correct answer)

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function M(t) = 1000(0.80)^t, the base is 0.80. To find the percent rate, we calculate r = 0.80 - 1 = -0.20. Converting to percent: -0.20 × 100% = -20%. Since 0.80 is less than 1, this is decay, specifically 20% decay per hour. Choice D correctly identifies the percent rate as 20% decay by showing correct reasoning. Excellent! Choice B gives the growth factor (b = 0.80) when the question asks for the growth rate (r = 20% decay). Remember: factor is what you multiply by, rate is the percent change. They're related by b = 1 + r! The form y = a(1 + r)^x makes the rate super obvious: if you see y = 500(1 + 0.08)^t, you can read the rate right off—it's 0.08 = 8%. But if you see y = 500(1.08)^t, you have to subtract 1 from the base: 1.08 - 1 = 0.08 = 8%. Same rate, just written differently!

Question 6

1. A 103% decrease for each 1-unit increase in $x$
2. A constant change of 3 units per 1-unit increase in $x$
3. A 3% increase for each 1-unit increase in $x$ (correct answer)
4. An initial value of 1.03

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.03), while the rate r is how much it's changing by percent (like 3%). They're related by b = 1 + r, so knowing one gives you the other! For the function y = 800·(1.03)^x, the base is 1.03. To find the percent rate, we calculate r = 1.03 - 1 = 0.03 as decimal. Converting to percent: 0.03 × 100% = 3%. Since 1.03 is greater than 1, this is growth, specifically 3% growth per unit in x. Choice B correctly identifies the percent rate as 3% by showing correct reasoning. Excellent! Choice C makes a common percent mistake: the base 1.03 doesn't mean 103% growth—it means 3% growth! The 1 represents 'what you already have' (100%), and the 0.03 is the additional 3%, for a total of 103% of the previous amount (which is 3% growth). Don't confuse the factor with the rate: if something grows by 5% per year, the growth RATE is 5% (r = 0.05), but the growth FACTOR is 1.05 (b = 1.05). Each year you have 105% of what you had (100% + 5%), which means multiplying by 1.05.

Question 7

1. 1% per month (correct answer)
2. 12% per month
3. 1% per year
4. 0.01% per month

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. Looking at the function A(t) = 1000(1.01)^{12t}, notice that the exponent is 12t, not just t. Since t is in years and the exponent includes 12t, this means we're compounding 12 times per year (monthly). The base 1.01 applies to each monthly period, so r = 1.01 - 1 = 0.01 = 1% per month. Choice A correctly identifies the monthly interest rate as 1% per month by recognizing that the base 1.01 applies to each of the 12 compounding periods per year. Excellent! Choice B might think that since there are 12 months, the rate is 12%, but that would be the approximate annual rate, not the monthly rate. The base 1.01 tells us each month the balance is multiplied by 1.01, which is 1% monthly growth. The form y = a(1 + r)^{nt} is common for compound interest, where n is the number of times per year interest is compounded. Here, with A(t) = 1000(1.01)^{12t}, we can see n = 12 (monthly) and the monthly rate is 1% since the base is 1.01.

Question 8

1. 0.10% depreciation
2. 10% depreciation (correct answer)
3. 10% growth
4. 90% depreciation

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function V(t) = 1200(0.90)^t, the base is 0.90. To find the percent rate, we calculate r = 0.90 - 1 = -0.10. Converting to percent: -0.10 × 100% = -10%. Since 0.90 is less than 1, this is decay, specifically 10% decay (depreciation) per year. Choice B correctly identifies the percent rate as 10% depreciation by calculating 1 - 0.90 = 0.10 = 10% decrease per year. Excellent! Choice A gives the growth factor (b = 0.90) when the question asks for the depreciation rate. Remember: if the base is 0.90, that means you keep 90% of the value each year, which is a 10% loss, not a 90% loss! Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form!

Question 9

1. 12% decay (correct answer)
2. 12% growth
3. 0.12% decay
4. 88% decay

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function y = 400(1 - 0.12)^x, we first simplify: 1 - 0.12 = 0.88, so y = 400(0.88)^x. The base is 0.88. To find the percent rate, we calculate r = 0.88 - 1 = -0.12. Converting to percent: -0.12 × 100% = -12%. Since 0.88 is less than 1, this is decay, specifically 12% decay per unit x. Choice A correctly identifies the percent rate as 12% decay by recognizing that (1 - 0.12) = 0.88 represents keeping 88% of the quantity, which is a 12% decrease. Excellent! Choice D gives the wrong rate: the base 0.88 means we keep 88% each time, which is a 12% loss, not an 88% loss! Remember to subtract from 1 to find the decay rate. The form y = a(1 - r)^x makes the decay rate super obvious: if you see y = 500(1 - 0.12)^t, you can read the rate right off—it's 0.12 = 12% decay. This form explicitly shows what percent is being lost each time period!

Question 10

1. 1.05% growth per year
2. 5% growth per year (correct answer)
3. 105% growth per year
4. 0.05% growth per year

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function A(t) = 1000(1.05)^t, the base is 1.05. To find the percent rate, we calculate r = 1.05 - 1 = 0.05. Converting to percent: 0.05 × 100% = 5%. Since 1.05 is greater than 1, this is growth, specifically 5% growth per year. Choice B correctly identifies the percent rate as 5% growth per year by showing that 1.05 = 1 + 0.05, which represents 5% growth. Excellent! Choice C makes a common percent mistake: the base 1.05 doesn't mean 105% growth—it means 5% growth! The 1 represents 'what you already have' (100%), and the 0.05 is the additional 5%, for a total of 105% of the previous amount (which is 5% growth). To find the percent rate: (1) Identify the base b = 1.05, (2) Subtract 1: r = 1.05 - 1 = 0.05, (3) Convert to percent: 0.05 × 100 = 5%. Example: base is 1.05, so r = 1.05 - 1 = 0.05 = 5%. Easy!

Question 11

1. $P_1$ (105% per week)
2. $P_2$ (10% per week) (correct answer)
3. They have the same growth rate
4. $P_1$ (5% per week)

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For P₁(t) = 100(1.05)^t, the base is 1.05, so r = 1.05 - 1 = 0.05 = 5% growth per week. For P₂(t) = 100(1.10)^t, the base is 1.10, so r = 1.10 - 1 = 0.10 = 10% growth per week. Since 10% > 5%, population P₂ has the higher weekly growth rate. Choice B correctly identifies P₂ as having the higher growth rate of 10% per week compared to P₁'s 5% per week. Excellent! Choice D makes a common percent mistake: the base 1.05 doesn't mean 105% growth—it means 5% growth! The 1 represents 'what you already have' (100%), and the 0.05 is the additional 5%, for a total of 105% of the previous amount (which is 5% growth). Real-world clue: when comparing exponential functions with the same initial value, just compare the bases—the larger base means faster growth. Here, 1.10 > 1.05, so P₂ grows faster than P₁.

Question 12

1. 3% decay
2. 0.03% growth
3. 103% growth
4. 3% growth (correct answer)

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function P(t) = 800(1.03)^t, the base is 1.03. To find the percent rate, we calculate r = 1.03 - 1 = 0.03. Converting to percent: 0.03 × 100% = 3%. Since 1.03 is greater than 1, this is growth, specifically 3% growth per year. Choice A correctly identifies the percent rate as 3% growth by showing correct reasoning. Excellent! Choice C makes a common percent mistake: the base 1.03 doesn't mean 103% growth—it means 3% growth! The 1 represents 'what you already have' (100%), and the 0.03 is the additional 3%, for a total of 103% of the previous amount (which is 3% growth). Don't confuse the factor with the rate: if something grows by 5% per year, the growth RATE is 5% (r = 0.05), but the growth FACTOR is 1.05 (b = 1.05). Each year you have 105% of what you had (100% + 5%), which means multiplying by 1.05.

Question 13

1. The growth factor is 5%
2. 105% growth per year
3. 5% growth per year (correct answer)
4. 0.05% growth per year

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function y = 1000·(1.05)^t, the base is 1.05. To find the percent rate, we calculate r = 1.05 - 1 = 0.05 as decimal. Converting to percent: 0.05 × 100% = 5%. Since 1.05 is greater than 1, this is growth, specifically 5% growth per year. Choice B correctly identifies the percent rate as 5% by showing correct reasoning. Excellent! Choice C makes a common percent mistake: the base 1.05 doesn't mean 105% growth—it means 5% growth! The 1 represents 'what you already have' (100%), and the 0.05 is the additional 5%, for a total of 105% of the previous amount (which is 5% growth). To find the percent rate: (1) Identify the base b, (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: base is 1.03, so r = 1.03 - 1 = 0.03 = 3%. For decay like 0.97: r = 0.97 - 1 = -0.03 = -3%, which we call '3% decay.' Easy!

Question 14

1. Linear decay
2. Exponential decay (correct answer)
3. Exponential growth
4. Neither (constant)

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.03), while the rate r is how much it's changing by percent (like 3%). They're related by b = 1 + r, so knowing one gives you the other! Looking at the function V(t) = 25000(0.88)^t, we check the base: 0.88 is less than 1, which means this is exponential decay. Think of it this way: each time t increases by 1, V is multiplied by 0.88, so V is getting smaller—that's decay! Choice C correctly identifies this as exponential decay because the base 0.88 is less than 1, meaning the car's value decreases exponentially over time. Excellent! Choice A confuses growth with decay: since the base 0.88 is less than 1, this is decay, not growth. An easy way to remember: bases bigger than 1 mean growing, bases between 0 and 1 mean shrinking! Here's your growth/decay decision tree: (1) Look at the base b, (2) Is b > 1? That's growth. Is 0 < b < 1? That's decay. Is b = 1? No change. That's it! For example, 1.07 > 1 so growth, 0.94 < 1 so decay, 1.00 = 1 so constant.

Question 15

1. 8% growth per hour (correct answer)
2. 1.08% growth per hour
3. 108% growth per hour
4. 8% decay per hour

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. In an exponential function y = a·b^x, the base b tells you whether it's growth or decay: if b > 1 (bigger than 1), the function is growing exponentially; if 0 < b < 1 (between 0 and 1), it's decaying. The initial value a is what you start with when x = 0. For the function P(t) = 500(1.08)^t, the base is 1.08. To find the percent rate, we calculate r = 1.08 - 1 = 0.08. Converting to percent: 0.08 × 100% = 8%. Since 1.08 is greater than 1, this is growth, specifically 8% growth per hour. Choice A correctly identifies the percent rate as 8% growth per hour by showing correct reasoning. Excellent! Choice B makes a common percent mistake: the base 1.08 doesn't mean 1.08% growth—it means 8% growth! The 1 represents 'what you already have' (100%), and the 0.08 is the additional 8%, for a total of 108% of the previous amount (which is 8% growth). To find the percent rate: (1) Identify the base b, (2) Subtract 1: r = b - 1, (3) Convert to percent: multiply by 100. Example: base is 1.08, so r = 1.08 - 1 = 0.08 = 8%. For decay like 0.97: r = 0.97 - 1 = -0.03 = -3%, which we call '3% decay.' Easy!

Question 16

1. An initial value of 1.10
2. A growth factor of 1.10 (10% increase each time $x$ increases by 1) (correct answer)
3. A decay rate of 10% each time $x$ increases by 1
4. A constant rate of change of 1.10

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.03), while the rate r is how much it's changing by percent (like 3%). They're related by b = 1 + r, so knowing one gives you the other! Looking at the function y = 200·(1.10)^x, we check the base: 1.10 is greater than 1, which means this is exponential growth. Think of it this way: each time x increases by 1, y is multiplied by 1.10, so y is getting bigger—that's growth! Choice B correctly identifies this as a growth factor of 1.10, representing a 10% increase each time x increases by 1. Excellent! Choice A confuses the base with the initial value: the initial value is 200 (when x = 0), not 1.10. The base 1.10 is what we multiply by each time x increases. Don't confuse the factor with the rate: if something grows by 10% per unit, the growth RATE is 10% (r = 0.10), but the growth FACTOR is 1.10 (b = 1.10). Each time you have 110% of what you had (100% + 10%), which means multiplying by 1.10.

Question 17

1. $b=0.10$
2. $b=1.10$ (correct answer)
3. $b=10$
4. $b=0.90$

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. The key difference between growth factor and growth rate: the factor b is what you multiply by each time (like 1.10), while the rate r is how much it's changing by percent (like 10%). They're related by b = 1 + r, so knowing one gives you the other! The context tells us the population increases by 10% each year, so the rate is r = 10% = 0.10. Converting the rate to the growth factor: b = 1 + r = 1 + 0.10 = 1.10. So the growth factor is 1.10. We can think of it this way: each year the population is 110% of what it was (100% + 10% increase). Choice B correctly identifies b = 1.10 by using the relationship b = 1 + r where r = 0.10. Excellent! Choice A gives the growth rate (r = 0.10) when the question asks for the growth factor (b = 1.10). Remember: factor is what you multiply by, rate is the percent change. They're related by b = 1 + r! Don't confuse the factor with the rate: if something grows by 10% per year, the growth RATE is 10% (r = 0.10), but the growth FACTOR is 1.10 (b = 1.10). Each year you have 110% of what you had (100% + 10%), which means multiplying by 1.10.

Question 18

1. 2% per month (correct answer)
2. 12% per month
3. 24% per month
4. 2% per year

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function A(t) = 1500(1.02)^{12t}, the base is 1.02 and it's applied each month (12 times per year). To find the monthly percent rate, we calculate r = 1.02 - 1 = 0.02. Converting to percent: 0.02 × 100% = 2%. Since 1.02 is greater than 1, this is growth, specifically 2% growth per month. Choice A correctly identifies the monthly interest rate as 2% per month by showing correct reasoning. Excellent! Choice C might be thinking about the annual rate (roughly 24% per year with compounding), but the question specifically asks for the monthly rate, which is 2%. The exponent 12t tells us the base is applied 12 times per year, once each month. Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form!

Question 19

1. 24% per month
2. 12% per month
3. 2% per month (correct answer)
4. 2% per year

Explanation: This question tests your understanding of exponential functions and how to identify whether they represent growth or decay and what the percent rate of change is. To find the percent growth or decay rate from the base, use the formula r = b - 1 and convert to percent: if b = 1.05, then r = 1.05 - 1 = 0.05 = 5% growth. If b = 0.95, then r = 0.95 - 1 = -0.05 = 5% decay (we usually just say '5% decay' and understand it's a decrease). For the function A(t) = 1500(1.02)^{12t}, the base is 1.02 and it's applied each month (12 times per year). To find the monthly percent rate, we calculate r = 1.02 - 1 = 0.02. Converting to percent: 0.02 × 100% = 2%. Since 1.02 is greater than 1, this is growth, specifically 2% growth per month. Choice A correctly identifies the monthly interest rate as 2% per month by showing correct reasoning. Excellent! Choice C might be thinking about the annual rate (roughly 24% per year with compounding), but the question specifically asks for the monthly rate, which is 2%. The exponent 12t tells us the base is applied 12 times per year, once each month. Real-world clue: 'percent interest' or 'percent increase' means exponential growth with that as your r. 'Percent depreciation' or 'percent decrease' means exponential decay. The problem language often tells you what type and what rate directly—you just translate to mathematical form!

Question 20

1. 24 years; annual decay rate of approximately 5.61% (correct answer)
2. 6 years; annual decay rate of approximately 50%
3. 24 years; annual decay rate of approximately 12.5%
4. 48 years; annual decay rate of approximately 94.4%

Explanation: To find when $N(t) = 0.25N_0$: $0.25 = (0.5)^{t/12}$, so $0.5^2 = (0.5)^{t/12}$, giving $t/12 = 2$, thus $t = 24$ years. The annual decay rate is $1 - (0.5)^{1/12} = 1 - 0.9439 = 5.61\%$. Choice B confuses the half-life period with the quarter-life. Choice C miscalculates the annual rate. Choice D doubles the time and inverts the decay calculation.