In the Player’s Handbook: Core Rulebook I v.3.5 on page 64 the Table 4–3: Difficulty Class Examples:
\begin{array}{lll}
\textbf{Difficulty (DC)} &
\textbf{Example (Skill Used)} \\
\hline
\text{Very easy (0)} &
\text{Notice something large in plain sight (Spot)} \\
\text{Easy (5)} &
\text{Climb a knotted rope (Climb)} \\
\text{Average (10)} &
\text{Hear an approaching guard (Listen)} \\
\text{Tough (15)} &
\text{Rig a wagon wheel to fall off (Disable Device)} \\
\text{Challenging (20)} &
\text{Swim in stormy waters (Swim)} \\
\text{Formidable (25)} &
\text{Open an average lock (Open Lock)} \\
\text{Heroic (30)} &
\text{Leap across a 30-foot chasm (Jump)} \\
\text{Nearly impossible (40)} &
\text{Track a squad of orcs across hard ground after 24 hours of rainfall (Survival)} \\
\end{array}
Compare this table to the Typical Difficulty Classes table found in the fifth edition Player’s Handbook on page 174:
\begin{array}{lll}
\textbf{Task Difficulty} &
\textbf{DC} \\
\hline
\text{Very easy} & \text{5} \\
\text{Easy} & \text{10} \\
\text{Medium} & \text{15} \\
\text{Hard} & \text{20} \\
\text{Very hard} & \text{25} \\
\text{Nearly impossible} & \text{30} \\
\end{array}
You can see the task difficulties of Very easy, Easy, and Nearly impossible exist in both. So a possible formula to convert from a third edition DC to fifth edition is:
If the 3E DC is between 0 and 5, just add 5 to create the equivalent 5E DC.
After the Easy task difficulty the labels don't match up again until Nearly impossible. We have the range of values in 3E from 5–40 that need to map onto the 5E range of 10–30. Therefore, each 5E DC will be 4/7 the equivalent 3E DC. Because the 3E DC will be greater than 5 we need to subtract 5 before multiplying by 4/7 and afterwards the 5E value needs to be greater than 10 so we'll add 10.
So if the 3E DC is greater than 5, a possible formula to convert to 5E DC would be:
5E DC = (3E DC - 5) * (4 / 7) + 10
In the question's examples the DC 30 Climb check would become:
(30 - 5) * (4 / 7) + 10 = ~24