Quick Play mode scores are the only ones that count toward this Achievement, not Brain Training. Click Profile/Awards to check your progress in each exercise.
Don't be afraid to pause by pressing the Back button if it might help with an exercise.
For the really unfair ones (Umbrellas, How Many Balls, etc.), consider recording the sequence before the question (using a camera, etc.), pausing, and then watching the video to determine the answer. Then unpause and quickly choose the right answer. This will allow you to get drastically better times.
You don't actually need to score 150 or higher in all games - the average of all scores just needs to be 150 or higher. This allows people like me to get by with a slightly lower Umbrellas score.