Exponential Growth and Decay Word Problems Riddle - Free Printable

Let’s solve each problem one by one. We’ll write the equation, plug in the numbers, calculate carefully, and round to the nearest whole number as instructed.

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Problem W: Sarah invests $500 at 1.5% compounded quarterly for 5 years.

Formula for compound interest:
A = P(1 + r/n)^(nt)

Where:
P = 500 (principal)
r = 1.5% = 0.015 (annual rate)
n = 4 (compounded quarterly)
t = 5 (years)

So:
A = 500 × (1 + 0.015/4)^(4×5)
= 500 × (1 + 0.00375)^20
= 500 × (1.00375)^20

Calculate (1.00375)^20:
≈ 1.0778 (using calculator)

Then:
A ≈ 500 × 1.0778 = 538.9 → Round to 539

Wait — but looking at the answer choices at the bottom: 22,756; 112; 12,800; 673; 30,425; 13

539 is not among them. Did I misread? Let me double-check.

Actually — maybe I made a mistake. Let me recalculate (1.00375)^20 more accurately.

Using precise calculation:

1.00375^20 = e^(20 * ln(1.00375)) ≈ e^(20 * 0.003743) ≈ e^0.07486 ≈ 1.0777

Still ~538.85 → rounds to 539.

But 539 isn’t an option. Hmm. Maybe I need to check if the problem meant something else? Or perhaps I should look at other problems first.

Wait — let’s try Problem A next.

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Problem A: Joseph buys car for $26,500, depreciates 3% annually for 5 years.

Depreciation formula:
A = P(1 - r)^t

P = 26500
r = 0.03
t = 5

A = 26500 × (0.97)^5

Calculate (0.97)^5:

0.97^2 = 0.9409
0.97^4 = (0.9409)^2 ≈ 0.88529281
0.97^5 = 0.88529281 × 0.97 ≈ 0.858734

Then:
A ≈ 26500 × 0.858734 ≈ 22756.45 → Round to 22,756

That matches one of the answers! So A → 22,756

Okay, so my earlier calculation for W must be wrong because 539 isn't listed. Let me recheck W.

Wait — maybe I miscalculated the exponent or rate?

Sarah: $500, 1.5% compounded quarterly, 5 years.

r = 0.015, n=4, t=5 → nt=20

(1 + 0.015/4) = 1.00375

1.00375^20:

Let me compute step by step:

1.00375^2 = 1.0075140625
^4 = (1.0075140625)^2 ≈ 1.015084
^8 ≈ (1.015084)^2 ≈ 1.030395
^16 ≈ (1.030395)^2 ≈ 1.061714

Now ^20 = ^16 × ^4 ≈ 1.061714 × 1.015084 ≈ 1.0777 → same as before.

500 × 1.0777 = 538.85 → still 539.

But 539 not in options. Unless... did I misread the problem?

Wait — maybe it's simple interest? No, it says “compounded quarterly”.

Alternatively — perhaps the answer key has a typo? But let’s keep going.

Maybe I should do all problems and see which ones match.

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Problem L: Medication metabolizes at 4% per hour. Take 20mg at 8am, how much at 6pm?

From 8am to 6pm is 10 hours.

Decay formula: A = P(1 - r)^t

P = 20
r = 0.04
t = 10

A = 20 × (0.96)^10

Calculate (0.96)^10:

0.96^2 = 0.9216
0.96^4 = (0.9216)^2 ≈ 0.84934656
0.96^8 ≈ (0.84934656)^2 ≈ 0.721389
Then ^10 = ^8 × ^2 ≈ 0.721389 × 0.9216 ≈ 0.6648

So A ≈ 20 × 0.6648 = 13.296 → Round to 13

That’s one of the options! So L → 13

Good.

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Problem E: Population 22,640 in 2015, grows 3% yearly, what in 2025?

2025 - 2015 = 10 years

Growth formula: A = P(1 + r)^t

P = 22640
r = 0.03
t = 10

A = 22640 × (1.03)^10

(1.03)^10 ≈ 1.343916 (standard value)

A ≈ 22640 × 1.343916 ≈ ?

First, 22640 × 1.3 = 29432
22640 × 0.043916 ≈ let’s compute:

22640 × 0.04 = 905.6
22640 × 0.003916 ≈ 22640 × 0.004 = 90.56 minus a bit → say 88.6

So total extra ≈ 905.6 + 88.6 = 994.2

Total A ≈ 29432 + 994.2 = 30426.2 → Round to 30,425? Wait, 30426.2 rounds to 30426, but option is 30,425.

Close enough — probably rounding difference.

Let me calculate exactly:

22640 × 1.343916 = ?

22640 × 1.343916

Break it down:

22640 × 1 = 22640
22640 × 0.3 = 6792
22640 × 0.04 = 905.6
22640 × 0.003 = 67.92
22640 × 0.0009 = 20.376
22640 × 0.000016 = 0.36224

Add up:

22640
+6792 = 29432
+905.6 = 30337.6
+67.92 = 30405.52
+20.376 = 30425.896
+0.36224 ≈ 30426.258

So ≈ 30426 → but the option is 30,425. Maybe they used slightly different rounding.

Perhaps (1.03)^10 is taken as 1.3439, then 22640 × 1.3439 = ?

22640 × 1.3439

= 22640 × (1.3 + 0.0439) = 22640×1.3 = 29432; 22640×0.0439

22640 × 0.04 = 905.6
22640 × 0.0039 = 88.296
Total 905.6 + 88.296 = 993.896

Total A = 29432 + 993.896 = 30425.896 → rounds to 30,426, but option is 30,425.

Hmm. Perhaps they expect us to use 1.3439 and get 30425.896 and round down? Or maybe a calculation error.

Looking at options: 30,425 is there. Probably acceptable. So E → 30,425

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Problem T: Computer bought for $870 in 2021, depreciates 40% per year, worth in 2025?

2025 - 2021 = 4 years

Depreciation: A = P(1 - r)^t

P = 870
r = 0.40
t = 4

A = 870 × (0.60)^4

(0.6)^2 = 0.36
(0.6)^4 = 0.36^2 = 0.1296

A = 870 × 0.1296 = ?

870 × 0.13 = 113.1
But 0.1296 is 0.0004 less → 870 × 0.0004 = 0.348
So 113.1 - 0.348 = 112.752 → Round to 113?

But option is 112.

Compute exactly:

870 × 0.1296

800 × 0.1296 = 103.68
70 × 0.1296 = 9.072
Total = 103.68 + 9.072 = 112.752 → rounds to 113

But option is 112. Hmm.

Perhaps they want floor? Or maybe I miscalculated.

(0.6)^4 = 0.1296 exactly.

870 × 0.1296:

Let me do 870 × 1296 / 10000

870 × 1296 = ?

800×1296 = 1,036,800
70×1296 = 90,720
Total = 1,127,520

Divide by 10000: 112.752 → yes.

Rounds to 113, but option is 112. Maybe typo? Or perhaps they expect truncation? But instruction says "round to nearest whole number".

112.752 is closer to 113 than 112.

But let’s see the options: 112 is there. Perhaps for this problem, we go with 112? Or maybe I have a mistake.

Wait — depreciation: is it 40% of original each year? No, typically it's exponential decay, which I did.

Perhaps they mean linear? But no, it says "depreciate at a rate of about 40% per year" — usually means exponential.

But let’s assume it’s correct and move on. Maybe later we can adjust.

T → 112? Even though math says 113. Perhaps in context, we take 112.

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Problem O: Bacteria start with 200, double every 4 hours. How many at end of day?

End of day = 24 hours.

Doubling every 4 hours → number of doublings = 24 / 4 = 6

So A = 200 × 2^6 = 200 × 64 = 12,800

Exactly! And 12,800 is an option.

So O → 12,800

Perfect.

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Now back to W: Sarah’s investment.

We got 539, but not in options. Options are: 22,756; 112; 12,800; 673; 30,425; 13

673 is there. Did I misread the problem?

"Sarah invests $500 at 1.5% compounded quarterly. After 5 years."

Perhaps it's 15%? But no, it says 1.5%.

Or maybe time is different? 5 years.

Another thought: perhaps "compounded quarterly" but they mean something else? No.

Let me calculate again with more precision.

A = 500 * (1 + 0.015/4)^(20) = 500 * (1.00375)^20

Use calculator: 1.00375^20 = 1.077831...

500 * 1.077831 = 538.9155 → 539

Not 673.

Unless... is it simple interest? 500 * 0.015 * 5 = 37.5, total 537.5 → still 538.

No.

Perhaps the rate is annual but compounded differently? No.

Another idea: maybe "1.5%" is monthly? But it says "rate of 1.5% compounded quarterly" — usually that means annual rate.

In finance, when they say "1.5% compounded quarterly", it means annual nominal rate is 1.5%, so quarterly rate is 1.5%/4.

I think my calculation is correct.

But 673 is an option. What if it was 5% instead of 1.5%?

Try r=0.05: A = 500*(1+0.05/4)^20 = 500*(1.0125)^20

1.0125^20 ≈ 1.282037

500*1.282037 = 641.0185 → 641, not 673.

If r=6%: 0.06/4=0.015, (1.015)^20 = 1.346855, 500*1.346855=673.4275 → 673!

Oh! Probably a typo in the problem or in my reading. It says "1.5%", but if it were 6%, it would give 673.

But it clearly says "1.5%". However, since 673 is an option and matches 6%, perhaps it's a common mistake or typo.

Maybe in some contexts, but I think for the sake of matching, W might be intended to be 673, assuming rate is 6%.

But let's see the letters.

We have:

W: ?
A: 22,756
L: 13
E: 30,425
T: 112 (even though calc says 113, but option is 112)
O: 12,800

And options left: 673

So probably W is 673, implying the rate is 6%, not 1.5%. Perhaps a misprint.

Maybe I should proceed with that.

So assign:

W → 673
A → 22,756
L → 13
E → 30,425
T → 112
O → 12,800

Now, the riddle is "What gets wet while drying?" and we need to match letters to answers.

The answers at bottom are: 22,756; 112; 12,800; 673; 30,425; 13

Corresponding to:

A: 22,756
T: 112
O: 12,800
W: 673
E: 30,425
L: 13

So the word is formed by the letters corresponding to the answers in order? But the riddle is to find what gets wet while drying, and we have to match letter to answer to spell the word.

Typically in such worksheets, you solve each problem, get a number, find that number in the list at bottom, and the letter above it tells you which letter to put in the riddle.

But here, the riddle is given, and we need to find the answer by matching.

The instruction: "Match the letter in the box to the answers at the bottom of the page to answer the riddle"

So for each problem, we have a letter (W,A,L,E,T,O), and we solve to get a number, then find that number in the list at bottom, and the position or something? No.

Actually, usually, the answers at bottom are paired with letters, but here the letters are in the boxes.

Perhaps after solving, we get a number for each letter, then we look at the list at bottom, which has numbers, and we associate the letter with that number, but then how does that answer the riddle?

I think the standard way is: you solve each problem, get a numerical answer, then find that number in the list at the bottom, and the letter that is associated with that number (but in this case, the letters are in the boxes, not with the answers).

Looking back: "Match the letter in the box to the answers at the bottom"

Probably, after solving, for example, problem W gives 673, and 673 is in the list, so we know that W corresponds to 673, but then how does that help with the riddle?

Perhaps the riddle is answered by taking the letters in the order of the answers or something.

Another common format: the answers at bottom are to be matched to the problems, and then the letters spell out the answer to the riddle.

For example, if we have six problems, and six answers, we assign each answer to its problem's letter, then arrange the letters based on the order of answers or something.

But the riddle is "What gets wet while drying?" and the answer is likely "a towel".

So probably, the letters should spell "TOWEL" or something.

We have letters: W,A,L,E,T,O

If we arrange them to form "TOWEL", that would be T,O,W,E,L — but we have six letters, including A.

"TOWEL" is five letters, we have six problems.

Perhaps "A TOWEL" but that's six characters.

Letters: T,O,W,E,L,A — could be "A TOWEL" but usually it's just "towel".

Perhaps the answer is "towel", and we need to select the letters that spell it.

From our assignments:

T: 112
O: 12,800
W: 673
E: 30,425
L: 13
A: 22,756

The answers at bottom are listed as: 22,756; 112; 12,800; 673; 30,425; 13

Which correspond to A, T, O, W, E, L

So if we read the letters in the order of the answers listed: first answer 22,756 is A, second 112 is T, third 12,800 is O, fourth 673 is W, fifth 30,425 is E, sixth 13 is L.

So the sequence is A,T,O,W,E,L

Which spells "ATOWEL" — almost "A TOWEL", but without space.

Probably the answer is "a towel", and the riddle is solved by recognizing that.

To confirm, "what gets wet while drying?" — a towel, because when you dry yourself with a towel, it gets wet.

So the final answer to the riddle is "a towel", but since it's a math worksheet, probably they want the word formed, which is "ATOWEL" or just "towel".

But in the context, since we have to box the answer, and the riddle is to be answered, likely "towel".

But let's see the instruction: "answer the riddle", so probably "towel".

However, in the matching, we have the letters, and when arranged by the answer order, we get A,T,O,W,E,L which is "atowel", suggesting "a towel".

Perhaps for the purpose, we can say the answer is "towel".

But to be precise, since the problem asks to match to answer the riddle, and the riddle is given, the expected answer is the word that solves it.

So I'll go with "towel".

But let's make sure all calculations are consistent.

Earlier for T, I had 112.752, rounded to 113, but option is 112. Perhaps they expect 112.

Similarly for E, 30426 vs 30425.

But for W, if we assume rate is 6%, then A=500*(1+0.06/4)^20=500*(1.015)^20

1.015^20 = (1.015^2)^10 = 1.030225^10

1.030225^2 = 1.061363550625
^4 = (1.061363550625)^2 ≈ 1.126492
^8 ≈ 1.268986
Then ^10 = ^8 * ^2 ≈ 1.268986 * 1.06136355 ≈ 1.346855 (as before)

500 * 1.346855 = 673.4275 → rounds to 673, perfect.

So likely, the rate is meant to be 6%, not 1.5%. Probably a typo in the problem.

Similarly, for T: computer depreciation.

A = 870 * (0.6)^4 = 870 * 0.1296 = 112.752

If they round down or something, but 112.752 is closer to 113, but option is 112. Perhaps in some systems, they truncate, but instruction says "round to nearest whole number".

112.752 - 112 = 0.752, 113 - 112.752 = 0.248, so actually closer to 113? No:

Distance to 112 is 0.752, to 113 is 0.248, so yes, closer to 113.

But 113 is not an option; only 112 is.

Unless I miscalculated the years.

Bought in 2021, worth in 2025.

2021 to 2022: 1 year
2022 to 2023: 2
2023 to 2024: 3
2024 to 2025: 4 years. Yes.

Perhaps depreciation is applied at the end of each year, but still same.

Maybe "depreciate at a rate of 40% per year" means something else, but I think exponential is correct.

Perhaps they mean linear depreciation: 40% of original per year.

Then after 4 years, depreciated by 4*40% = 160%, which is impossible.

So must be exponential.

Perhaps the rate is 40% of remaining, which is what I did.

Another thought: maybe "in 2025" means after 4 full years, but from 2021 to 2025 is 4 years, yes.

Perhaps the answer is 112, and we accept it.

Similarly for E, 30425.896, if they round to nearest, it's 30426, but option is 30425. Perhaps they used (1.03)^10 = 1.3439, and 22640*1.3439=30425.896, and rounded down? But usually round half up.

30425.896 is closer to 30426.

But since 30425 is the option, and it's very close, probably acceptable.

So to summarize:

- W: 673 (assuming rate is 6%)
- A: 22,756
- L: 13
- E: 30,425
- T: 112
- O: 12,800

And the letters in the order of the answers listed at bottom: 22,756 (A), 112 (T), 12,800 (O), 673 (W), 30,425 (E), 13 (L) → A,T,O,W,E,L → "ATOWEL" → "a towel"

So the answer to the riddle is "a towel", but typically just "towel".

Since the riddle is "What gets wet while drying?", the answer is "a towel".

For the boxed answer, probably "towel".

To confirm, let's see if there's another interpretation.

Perhaps the matching is to find which letter corresponds to which answer, and then the riddle is solved by the word formed by the letters in alphabetical order or something, but that doesn't make sense.

I think "towel" is correct.

So final answer for the riddle is "towel".

But the user asked to solve the problem, and the problem includes solving each part and answering the riddle.

Since the riddle is the main goal, and we've done the math, I'll box "towel".

But to be thorough, let's list all answers:

Problem W: 673
Problem A: 22,756
Problem L: 13
Problem E: 30,425
Problem T: 112
Problem O: 12,800

And the riddle answer is "towel".

So for the Final Answer, since the riddle is to be answered, I'll put "towel".

Perhaps they want the word from the letters, but I think "towel" is fine.

Another way: sometimes the answer is the word spelled by the letters corresponding to the solutions in order, but here the solutions are not ordered.

I think it's safe.

So Final Answer: towel

But let's write it as per instruction.