Copy paste specific rows from one table to another table based on matching value

Hi novicepowerbi 
You can get the desired result by those steps :
1. merge details with summary table, by activity id :

after this step, you will have just needed rows from the details table, so you can delete the newly created column

next step is just to append the tables

result:

Note that you cant delete unnecessary table because of merge , so you can just not to load it to the model

Pbix is attached

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Based on your description, I created data to reproduce your scenario. The pbix file is attached in the end.