In this question, we are focussed only on the final year students in AMAS university in a certain year. You are given 75% of all these students graduated, and that 60% graduated among the female students. Which of the following statements must be true? Select all that are true.Group of answer choicesThe percentage of male students among the graduates is less than the percentage of male students among the non-graduates.The percentage of graduates among the female students is less than the percentage of graduates among the male students.The percentage of male students is greater than the percentage of male students among the non-graduates.The percentage of female students among the non-graduates is greater than the percentage of female students among the graduates.

Let's analyze the given information step by step:

1. 75% of all final year students graduated.
2. 60% of the female students graduated.

We need to determine which of the following statements must be true:

1. The percentage of male students among the graduates is less than the percentage of male students among the non-graduates.
2. The percentage of graduates among the female students is less than the percentage of graduates among the male students.
3. The percentage of male students is greater than the percentage of male students among the non-graduates.
4. The percentage of female students among the non-graduates is greater than the percentage of female students among the graduates.

Let's denote:

• $F$ as the total number of female students.
• $M$ as the total number of male students.
• $T = F + M$ as the total number of final year students.

From the given information:

• 75% of $T$ graduated, so $0.75T$ students graduated.
• 60% of $F$ graduated, so $0.60F$ female students graduated.

The number of male graduates can be calculated as: $\text{Male graduates} = 0.75T - 0.60F$

The number of non-graduates is: $\text{Non-graduates} = T - 0.75T = 0.25T$

The number of female non-graduates is: $\text{Female non-graduates} = F - 0.60F = 0.40F$

The number of male non-graduates is: $\text{Male non-graduates} = 0.25T - 0.40F$

Now, let's evaluate each statement:

1. The percentage of male students among the graduates is less than the percentage of male students among the non-graduates.

   • Percentage of male students among graduates: $\frac{0.75T - 0.60F}{0.75T}$
   • Percentage of male students among non-graduates: $\frac{0.25T - 0.40F}{0.25T}$
   • Since $\frac{0.75T - 0.60F}{0.75T} < \frac{0.25T - 0.40F}{0.25T}$, this statement is true.

2. The percentage of graduates among the female students is less than the percentage of graduates among the male students.

   • Percentage of graduates among female students: $0.60$
   • Percentage of graduates among male students: $\frac{0.75T - 0.60F}{M}$
   • Since $0.60 < \frac{0.75T - 0.60F}{M}$, this statement is true.

3. The percentage of male students is greater than the percentage of male students among the non-graduates.

   • Percentage of male students: $\frac{M}{T}$
   • Percentage of male students among non-graduates: $\frac{0.25T - 0.40F}{0.25T}$
   • Since $\frac{M}{T} > \frac{0.25T - 0.40F}{0.25T}$, this statement is true.

4. The percentage of female students among the non-graduates is greater than the percentage of female students among the graduates.

   • Percentage of female students among non-graduates: $\frac{0.40F}{0.25T}$
   • Percentage of female students among graduates: $\frac{0.60F}{0.75T}$
   • Since $\frac{0.40F}{0.25T} > \frac{0.60F}{0.75T}$, this statement is true.

Therefore, all the given statements are true.