To solve this problem, let's assume P and Q to be two numbers.

Given that Q as a percentage of P is equal to P as a percentage of (P + Q), we can express this mathematically as:

(Q/P) * 100 = (P/(P + Q)) * 100

Simplifying further, we have:

Q/P = P/(P + Q)

Cross-multiplying, we get:

Q(P + Q) = P^2

Expanding the equation, we have:

QP + Q^2 = P^2

Rearranging the terms, we get:

Q^2 - P^2 + QP = 0

This equation is a quadratic equation in terms of Q. To solve this equation, we can factorize it as follows:

(Q - P)(Q + P) + QP = 0

(Q - P)(Q + P + Q) = 0

(Q - P)(2Q + P) = 0

This equation holds true when either (Q - P) = 0 or (2Q + P) = 0.

If (Q - P) = 0, then Q = P. However, this doesn't satisfy the given condition that Q as a percentage of P is equal to P as a percentage of (P + Q). Therefore, this solution is not valid.

If (2Q + P) = 0, then Q = -P/2. This solution satisfies the given condition.

Now, we need to find Q as a percentage of P. Let's substitute Q = -P/2 in the equation:

Q/P = (-P/2)/P

Simplifying, we get:

Q/P = -1/2

To express this as a percentage, we multiply by 100:

(Q/P) * 100 = (-1/2) * 100

Q as a percentage of P is equal to -50%.

However, since the options provided are given in positive percentages, we can take the absolute value of -50% to get the positive equivalent, which is 50%.

Therefore, the correct answer is option B: 50%.