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3 Responses
Great question! What you’re missing is that with compounding, each year’s 5% is on the new, bigger balance, so you multiply 1000×1.05×1.05×1.05 = 1157.625, not 1000×1.15 = 1150-your method is simple interest on the original only. Think of a snowball: each roll adds snow to the snow you already added, so the gains themselves start earning gains.
I used to trip on this too-“5% for three years is 15%” sounds so reasonable! The catch is that compounding means each year’s 5% is taken on the new, bigger balance, not on the original $1000. Your step “year 2 = 1000 × 1.10” quietly resets the base to $1000, which is simple interest. With compounding, we keep multiplying by 1.05 because we’re applying the same percentage to a changing amount. Constant rate doesn’t mean constant add-on; it means constant growth factor.
Worked example: Simple interest would give 1000 × (1 + 0.05 × 3) = 1000 × 1.15 = $1150. Compounding does it step by step: year 1 = 1000 × 1.05 = 1050; year 2 = 1050 × 1.05 = 1102.50; year 3 = 1102.50 × 1.05 = 1157.625, which matches 1000 × (1.05)^3 = $1157.625. The extra $7.625 is “interest on interest.” Over longer periods that gap grows a lot. A nice refresher with visuals is here: https://www.khanacademy.org/math/algebra/x054d8f63:exponential-functions/x054d8f63:introduction-to-compound-interest/v/compound-interest-introduction
You’re mixing simple interest with compound interest. With compounding, each year’s 5% is applied to the current balance, not the original $1,000. So: year 1 gives 1,000 → 1,050; year 2 gives 1,050 → 1,102.50; year 3 gives 1,102.50 → 1,157.625. That’s 1,000 × (1.05)^3. Your steps “year 2 = 1,000 × 1.10” and “year 3 = 1,000 × 1.15” keep going back to the original principal, which is simple interest: 1,000 × (1 + 0.05 × 3) = 1,150.
The constant rate doesn’t mean you add the same dollar amount each year; it means you multiply by the same factor each year. That creates “interest on interest.” Numerically, (1.05)^3 = 1 + 3(0.05) + 3(0.05)^2 + (0.05)^3, so it’s the 15% you expected plus an extra 0.7625% from compounding, which is the $7.625 difference between $1,157.625 and $1,150.