The Equation at Work
Four problems: an inherited $3,000, a credit card spiral, working backwards from a savings goal using P = A / (1 + r)^t, and the counterintuitive story of two investors who start at different ages.
Work through each problem. Use the hint if you're stuck, and check your answers when you're done with each part.
You receive a gift of $3,000 at age 16. You have two realistic options: spend it now, or invest it in an index fund earning an average of 8% per year and leave it until age 65.
How much would the $3,000 be worth at age 65 if invested? (t = 49 years)
Show answer
A = 3,000 × (1.08)49 ≈ 3,000 × 43.4 ≈ $130,000.
(1.0810 ≈ 2.16, 1.0820 ≈ 4.66, 1.0840 ≈ 21.7, 1.0849 ≈ 43.4)
What if you wait just 5 extra years and invest at age 21 instead of 16? (t = 44 years)
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A = 3,000 × (1.08)44 ≈ 3,000 × 29.6 ≈ $88,700.
Waiting just 5 years costs about $41,000 — from the same initial $3,000 investment. That's the exponent at work.
Is there a "right" answer here? What factors might legitimately go into the decision?
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No single right answer. Relevant factors include: whether you have emergency savings, what you'd spend it on, whether 8% is realistic for your situation, and your values around spending vs. saving. The formula tells you the cost of choosing now over later — not whether that cost is worth it.
HintShowHide
Use A = P(1 + r)t. P = $3,000, r = 0.08. For part (a), t = 65 − 16 = 49 years. For part (b), t = 65 − 21 = 44 years. A calculator handles 1.0849 directly — try it.
You have a credit card balance of $1,800. The annual interest rate is 24%. You make no payments.
What's the balance after 3 years?
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A = 1,800 × (1.24)3 = 1,800 × 1.907 ≈ $3,432.
After 7 years?
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A = 1,800 × (1.24)7 ≈ 1,800 × 4.51 ≈ $8,118.
(1.244 ≈ 2.36, 1.247 ≈ 2.36 × 1.91 ≈ 4.51)
How many years does it take for the balance to double? How does the Rule of 72 compare?
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The exact answer is between 3 and 4 years: 1.243 ≈ 1.91 (not quite doubled), 1.244 ≈ 2.36 (past doubled). Actual doubling time ≈ 3.2 years.
Rule of 72: 72 ÷ 24 = 3 years — a reasonable approximation.
HintShowHide
Use A = P(1 + r)t with P = $1,800 and r = 0.24. For part (c), the Rule of 72 says: years to double ≈ 72 ÷ interest rate (as a whole number). Try 72 ÷ 24.
You want to have $50,000 saved by age 40. You're currently 20. You expect an annual return of 6%.
How much would you need to invest today as a single lump sum?
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P = 50,000 ÷ (1.06)20 = 50,000 ÷ 3.207 ≈ $15,590.
What if you waited until age 30 to invest the lump sum (t = 10 years)?
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P = 50,000 ÷ (1.06)10 = 50,000 ÷ 1.791 ≈ $27,920.
Waiting 10 years nearly doubles the required investment — $27,920 instead of $15,590.
Does that surprise you? What does it tell you about starting early?
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The difference ($12,330) is entirely the cost of 10 fewer years of compounding. It's not that the amount you need goes up — it's that you're starting with less time for the money to grow, so you need to put in more upfront to reach the same destination.
HintShowHide
Rearrange the formula: if A = P(1 + r)t, then P = A ÷ (1 + r)t. You want A = $50,000, r = 0.06, t = 20. Then try t = 10 for the second part.
Marisol, age 25, invests $300 per month for 10 years, then stops completely at age 35 — never investing another dollar. Theo, age 25, waits until he's 35, then invests $300 per month for 30 years until age 65. Both earn 7% annually.
Note: this problem involves regular monthly contributions, which uses a more advanced formula. Focus on the intuition — make a prediction first, then check the result.
Who do you predict ends up with more money at age 65? Why?
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Most people predict Theo — he invests for 30 years vs. Marisol's 10, and puts in $108,000 total vs. her $36,000. But the answer is Marisol. The prediction is a useful check on your intuition about compounding.
Using a financial calculator or the future-value-of-annuity formula, find the approximate result. Who wins?
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Marisol ends up with approximately $395,000 at age 65. Theo ends up with approximately $340,000.
Marisol wins — despite investing for only 10 years and contributing $72,000 less. Her money had an extra 30 years to compound after she stopped contributing.
What does this tell you about what matters most in long-term investing?
Show answer
Time matters more than amount. Marisol's money started growing 10 years earlier, and that head start compounded into a larger outcome despite far fewer total contributions. The exponent in A = P(1+r)t is time — and time is what you can't get back.
HintShowHide
Before calculating anything: make a prediction. Theo invests 3× as long and puts in 3× as much total money. Does that mean he ends up with more? Think about when each person's money starts compounding.