Compound interest versus simple interest

Statement

We invest $4000\ \text{€}$ at $5\%$ for $10$ years. Compare the accumulated value with annual compound interest and with simple interest.

Solution

Compound: $M = 4000\cdot 1{.}05^{10} = 4000\cdot 1{.}628895 \approx \boxed{6515{.}58\ \text{€}}$. Simple: $M = 4000\,(1 + 0{.}05\cdot 10) = 4000\cdot 1{.}5 = \boxed{6000\ \text{€}}$. Compounding yields about $515{.}58\ \text{€}$ more, because interest itself earns interest.