A solution of $100\text{L}$ contains $75\%$ water and the rest liquid sugar. How much liquid sugar must be added to make it a $50\%$ sugar solution?

Initial total volume = $100\text{L}$ Initial sugar = $100-75=25\%$ of $100\text{L}=25\text{L}$ Initial water = $75\text{L}$ Let $x$ litres of sugar be added.New total volume = $100+x$ New sugar volume = $25+x$ We want the new sugar concentration to be $50\%$:
$$\frac{25+x}{100+x}=\frac{50}{100}=\frac{1}{2}$$
$$2(25+x)=100+x$$
$$50+2x=100+x$$
$$x=50\text{L}$$