What proportion of female students are good ?

Concept: in a two-way table every row total is the sum of that row’s cells and every column total is the sum of that column’s cells, and the grand total is reached along both directions. Any statement that fixes a share of the whole — a percentage of all students, or a fraction of one row — can therefore be written as an equation in those totals. A table with missing cells is recovered by turning each stated share into an equation and then filling the cells in an order where only one unknown appears at a time.

Applying this to the recovered table:

1. The Female row total is 32 and females are 40% of all students, so 0.40 × (grand total) = 32, which gives a grand total of 80 students.

2. Male total = 80 − 32 = 48.

3. One-third of the male students are Average, so the Male–Average cell = 48 ÷ 3 = 16.

4. Half of all students are Excellent or Good, so Excellent + Good = 80 ÷ 2 = 40, and therefore the Average column total = 80 − 40 = 40.

5. Female–Average = 40 − 16 = 24.

6. The Good column total is 30, so the Excellent column total = 40 − 30 = 10.

7. The Male–Excellent cell is 10, so Female–Excellent = 10 − 10 = 0.

8. Female–Good = 32 − 24 − 0 = 8.

9. Proportion of female students who are good = 8 ÷ 32 = 0.25.

Cross-check — the fully reconstructed table:

Male–Good is obtained as 30 − 8 = 22, and every row and every column now adds to its own total: 16 + 22 + 10 = 48, 24 + 8 + 0 = 32, and 40 + 30 + 10 = 80. Excellent + Good = 10 + 30 = 40, which is exactly half of 80, and 32 out of 80 is 40% female, so all three stated facts hold. The required proportion is 8 out of 32, that is 0.25.