Let be x the initial number of cups of punch. Then, we can write in algebraic language the word problem: x/2 because her guests drank 1/2 of punch.
And we can write the equation:
[tex]x-\frac{x}{2}-20=10[/tex]Because there was an initial amount of cups of punch, and then this amount decreased until there were 10 cups of punch left.
Now, we can write the equation for x:
[tex]\begin{gathered} x-\frac{x}{20}-20=10 \\ \text{ Add 20 from both sides of the equation} \\ x-\frac{x}{2}-20+20=10+20 \\ x-\frac{x}{2}=30 \\ \frac{2x}{2}-\frac{x}{2}=30 \\ \frac{2x-x}{2}=30 \\ \frac{x}{2}=30 \\ \text{ Multiply by 2 from both sides} \\ \frac{x}{2}\cdot2=30\cdot2 \\ \boldsymbol{x=60} \end{gathered}[/tex]Therefore, the initial number of cups that she started with was 60.