12. (6 points) Three lathes $A, B, C$ process the same type of standard parts at certain efficiencies. Lathe $A$ starts 10 minutes earlier than lathe $C$, and lathe $C$ starts 5 minutes earlier than lathe $B$. 10 minutes after lathe $B$ starts, the number of standard parts processed by lathes $B$ and $C$ is the same. 30 minutes after lathe $C$ starts, the number of standard parts processed by lathes $A$ and $C$ is the same. How many minutes after lathe $B$ starts will the number of standard parts processed by lathe $B$ be the same as that processed by lathe $A$?
【Analysis】First, based on the principle that efficiency and time are inversely proportional when the workload is the same, find the time ratio to determine the efficiency ratio, and then the time can be calculated.

【Solution】Solution: According to the problem:
Machine $A$ starts 10 minutes before Machine $C$, and Machine $B$ starts 5 minutes after Machine $C$.
When Machine $B$ has been running for 10 minutes, Machine $C$ has been running for 15 minutes, the time ratio is: $2: 3$, so the efficiency ratio is 3: 2.
When Machine $C$ has been running for 30 minutes, Machine $A$ has been running for 40 minutes, the time ratio is 3: 4, the efficiency ratio is 4: 3.
The efficiency ratio in a continuous form is $A: B: C = 3: 6: 4$.
Given that the efficiency of Machine $B$ is twice that of Machine $A$. The time difference is 15 minutes, and after another 15 minutes, the amount of work done will be the same. Answer: Machine $B$ will have the same amount of work done as Machine $A$ 15 minutes after it starts.