Given: \(\frac{x}{x + y}*10 + \frac{y}{x + y}*20 = k\)

Rewrite as follows: \(\frac{10x}{x + y} + \frac{20y}{x + y} = k\)

Rewrite as follows: \((\frac{10x}{x + y} + \frac{10y}{x + y}) + \frac{10y}{x + y}= k\)

Simplify: \((\frac{10x + 10y}{x + y}) + \frac{10y}{x + y}= k\)

Simplify: \(10 + \frac{10y}{x + y}= k\)

Rewrite as follows: \(10 + 10(\frac{y}{x + y})= k\)

IMPORTANT: If \( x = y\), then \(\frac{y}{x + y} = \frac{y}{y + y} = \frac{y}{2y} = \frac{1}{2}\)

Since it's actually the case that \( x < y\), then we know that \(\frac{1}{2} < \frac{y}{x + y} < 1 \), which means \(10(\frac{y}{x + y})\) is greater than 5 but less than 10.

So, we know that \(15 < k < 20\)

Answer: D