If the income of P is 40% less than Q, then the combined income of P and Q is how much per cent more than the income of P?

The correct answer is Option 4: $166\frac{2}{3}$%.

Step 1: Assume a base value for Q's income

Let the income of Q be 100 units.

Step 2: Calculate P's income

The problem states that the income of P is 40% less than Q.

Income of P = 100 - (40% of 100) = 100 - 40 = 60 units.

Step 3: Calculate the combined income of P and Q

Combined income of P and Q = Income of P + Income of Q

Combined income = 60 + 100 = 160 units.

Step 4: Find how much the combined income is more than P's income

Difference = Combined income - Income of P

Difference = 160 - 60 = 100 units.

Step 5: Calculate the required percentage

We need to find how much per cent the combined income (160) is more than P's income (60).

Percentage = (Difference / Income of P) × 100

$\text{Percentage}=\frac{100}{60}\times 100=\frac{500}{3}$%

Converting $\frac{500}{3}$ into a mixed fraction:

500 ÷ 3 = 166 with a remainder of 2.

Therefore, the percentage is $166\frac{2}{3}$%.