A person invested \(\frac{1}{3}\) of his capital at \(7\%\); \(\frac{1}{4}\) of his capital at \(8\%\)…

Let the capital be C. The remainder invested at 10 percent is $1 - \frac13 - \frac14 = \frac{5}{12}$ of the capital. The income is $C\left(\frac13\times0.07 + \frac14\times0.08 + \frac{5}{12}\times0.10\right) = 0.085C$, so $0.085C = 561$ and $C = 6600$. A simple average of the three rates, about 8.33 percent, would ignore the different amounts invested at each rate.