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The following table gives data of some students. However, some data has been spoilt and cannot be read on the computer floppy It is also known that:

1. Half the students were either excellent or good.

2. 40% of the students were females

3. One-third of the male students were average. Based on the above answer the following questions.

How many students are both female and excellent?

1. $0$
2. $8$
3. $16$
4. $32$

Performance   Total
Average         Good     Excellent
Male     10
Female       32
Total   30

We know that Total female students = 32

$40\%$ of the total students were females.

$∴ \text{Total students} = \dfrac{32 \times 100}{40} = 80$

Performance   Total
Average         Good     Excellent
Male     10
Female       32
Total   30   $\color{blue}{80}$

$∴\text{Total Male students = (80-32) = 48}$

Performance   Total
Average         Good     Excellent
Male     10 $\color{blue}{48}$
Female       32
Total   30   $\color{blue}{80}$

$\dfrac{1}{3}^{rd} \text{of the male students =} \{(\dfrac{1}{3}\times 48) = 16\}\text{ were average}$

Performance   Total
Average         Good     Excellent
Male $\color{blue}{16}$   10 $\color{blue}{48}$
Female       32
Total   30   $\color{blue}{80}$

$∴\text{ Good male students = 48 - (16+10)= 22}$

$∴\text{ Good female students = 30 - 22 = 8}$

Performance   Total
Average         Good     Excellent
Male $\color{blue}{16}$ $\color{blue}{22}$ 10 $\color{blue}{48}$
Female   $\color{blue}{8}$   32
Total   30   $\color{blue}{80}$

Now, we know that half of the students i.e. $\dfrac{80}{2} = 40$ students were either excellent or good.

∴ Rest $(80-40)=40$ students were average

Performance   Total
Average         Good     Excellent
Male $\color{blue}{16}$ $\color{blue}{22}$ 10 $\color{blue}{48}$
Female $\color{blue}{(40-16) = 24}$ $\color{blue}{8}$   32
Total $\color{blue}{40}$ 30   $\color{blue}{80}$

∴ $\color{green}{\text{There are}}$ $\color{red}{no}$ $\color{green}{\text{female students who are excellent.}}$

Performance   Total
Average         Good     Excellent
Male $\color{blue}{16}$ $\color{blue}{22}$ 10 $\color{blue}{48}$
Female $\color{blue}{(40-16) = 24}$ $\color{blue}{8}$   -- 32
Total $\color{blue}{40}$ 30 $\color{blue}{10}$ $\color{blue}{80}$
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