Get quartiles and a half of the data

The following table shows a frequency distribution of the scores obtained in a test.
Punctuation (3, 4] (4, 5] (5, 6] (6, 7] (7, 8] (8, 9] (9, 10]
Number of participants 2 4 10 20 40 35 9

(a) The highest score reached by the bottom 20% of the participants and the score lowest obtained by the top 25% of the participants
(b) The quartiles of the distribution
For section a) to be continuous variables (grouped data) that can have any value within a range what I do is calculate the average for that 20%, that is to say who is in position 24. I calculate the relative frequency (that for this I need to know the absolute frequency) and multiplied it by the midpoint of the interval. I am applying according to the definition and the formula, but I am stuck.
It is a note that is between (6,7] and the average I get 0.1, so the note would be 6.1.
For b) of the quartiles I get that 1 is in (6,7], 2 is in (7,8] and 3 is in (8,9), but I do not know how to get the exact note
Can someone tell me if it is correct?