In a national science olympiad the rules state that each school may enter a maximum of three learners (participants). TABLE 6 below shows the relationship between the number of schools entering and the maximum number of participants.

From the problem statement, each school may enter a maximum of three learners (participants).
From the given data for 900 schools, there are 2700 participants.
The ratio of participants to schools is $\frac{2700}{900} = 3$.
So, Number of participants = Number of schools $\times 3$.

For A, Number of schools = 367.

For B, Number of participants = 15726.

The missing values are $\boxed{A = 1101, B = 5242}$.

8.1.2 Calculate the number of schools that entered the olympiad if a total of 32 712 people were involved on the day the olympiad was written.

Each school has 3 participants and 1 invigilating teacher.
Total people per school = $3 \text{ (participants)} + 1 \text{ (teacher)} = 4$ people.

Let $S$ be the number of schools.
Total people involved = $4 \times S$.
Given total people involved = 32712.

The number of schools that entered the olympiad is $\boxed{8178}$.

8.2.1 Calculate Matuli's median percentage mark.

Matuli's marks: 53, 48, 62, 80, 48, 58, 72, 48, 70, 86.
Ordered marks: 48, 48, 48, 53, 58, 62, 70, 72, 80, 86.

There are 10 marks (an even number). The median is the average of the two middle values (the 5th and 6th marks).
The 5th mark is 58.
The 6th mark is 62.

Matuli's median percentage mark is $\boxed{60\%}$.

8.2.2 Calculate Bianca's mean percentage mark.

Bianca's marks from the table are: 36, 42, 48, 58, 60, 61, 62, 76, 86.
The note states "Bianca's median percentage mark is 60%". If we order these 9 marks: 36, 42, 48, 58, 60, 61, 62, 76, 86, the 5th mark (median for 9 values) is 60, which matches the note. Thus, there are 9 marks for Bianca.

Bianca's mean percentage mark is $\boxed{58.78\%}$.

8.2.3 Determine the missing value C, the lower quartile mark, if Khotso's interquartile range (IQR) is 16.

Khotso's marks (already ordered as per note): 30, 47, C, 55, 60, 60, 68, 68, 70, 90.
There are 10 marks.
IQR = Upper quartile (Q3) - Lower quartile (Q1) = 16.

The upper quartile (Q3) is the median of the upper half of the data.
Upper half: 60, 68, 68, 70, 90.
Q3 is the 3rd value in this ordered upper half.

The lower quartile (Q1) is the median of the lower half of the data.
Lower half: 30, 47, C, 55, 60.
Q1 is the 3rd value in this ordered lower half, which is C.
We are given IQR = 16.

The missing value C, which is also the lower quartile mark, is $\boxed{52}$.

8.2.4 Bianca stated that Matuli performed better than she did in the practice tests. Give TWO possible reasons to support Bianca's statement.

Here are two reasons to support Bianca's statement:

• Matuli's mean percentage mark (62.5%) is higher than Bianca's mean percentage mark (58.78%). A higher mean indicates a better overall performance.
• Matuli's lowest mark is 48%, while Bianca's lowest mark is 36%. This shows that Matuli avoided very low scores and maintained a higher minimum performance level.