Step 1: Determine the Remainder Fraction

The man invests $\frac{1}{3}$ of his income and $\frac{1}{4}$ of his income. The remaining fraction of his income is calculated as $1 - \left(\frac{1}{3} + \frac{1}{4}\right)$.

$1 - \left(\frac{1}{3} + \frac{1}{4}\right) = 1 - \frac{7}{12} = \frac{5}{12}$

Step 2: Assume Total Income and Calculate Portions

Let the total income be ₹$1200$ (choosing a multiple of 3 and 4 to avoid fractions). The three parts invested are $P_1 = 400$, $P_2 = 300$, and $P_3 = 500$.

$P_1 = 400, \quad P_2 = 300, \quad P_3 = 500$

Step 3: Calculate Individual Simple Interests for 1 Year

Using the simple interest formula $\text{SI} = \frac{P \times R \times T}{100}$ for each part at rates $7\%$, $8\%$, and $10\%$ respectively:

$\text{SI}_1 = \frac{400 \times 7 \times 1}{100} = 28, \quad \text{SI}_2 = \frac{300 \times 8 \times 1}{100} = 24, \quad \text{SI}_3 = \frac{500 \times 10 \times 1}{100} = 50$

Step 4: Find Total Interest and Scale to Actual Value

The total calculated interest is $28 + 24 + 50 = 102$ units. Given that the actual total interest is ₹$561$, we find the multiplier: $\frac{561}{102} = 5.5$.

$\text{Actual Income} = 1200 \times 5.5 = 6600$