The average marks in Maths exam of students in group A, group B and both the groups together are 80, 90 and 87 respectively. Which of the following cannot be the total number of students in the two groups?

Let $a$ = number of students in Group A and $b$ = number of students in Group B.

Using the weighted average formula:

$\dfrac{80a + 90b}{a + b} = 87$

$80a + 90b = 87(a + b)$

$80a + 90b = 87a + 87b$

$90b - 87b = 87a - 80a$

$3b = 7a$

$\dfrac{a}{b} = \dfrac{3}{7}$

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Since $a : b = 3 : 7$, for some positive integer $k$:

$a = 3k$ and $b = 7k$

$a + b = 3k + 7k = 10k$

So the total number of students must be a multiple of $10$.

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$80 = 10 \times 8$ — possible

$20 = 10 \times 2$ — possible

$85 = 10 \times 8.5$ — not possible, since $k$ must be a whole number

Therefore, $85$ cannot be the total number of students.