How many family members are there?

Let $m$ be the total amount of milk and $c$ the total amount of coffee, and $N$ the number of family members.

About Ajay we know $\tfrac17 m+ \tfrac2{17}c = 8;\ \ \ \ m+c = 8\cdot N.$ We solve this system for equation for $m$ and $c$ , leaving $N$ as a variable (for now). I multiply the first equation by $7\cdot 17$ to remove the fractions. $\begin{array}{rrr} 17m + & 14c = & 952 \\ m + & c = & 8\cdot N \end{array}$ Multiply the second equation by 14 and subtract it from the first to solve for $m$ . Multiply the second equation by 17 and subtract the first from it to solve for $c$ . This gives $\begin{array}{c} 3m = 952-112\cdot N \\ 3c = 136\cdot N - 952\end{array}$ The only information we have yet is $m > 0$ and $c > 0$ . This means $N < \frac{952}{112} = 8\tfrac12;\ \ \ N > \frac{952}{136} = 7.$ The only integer solution to these inequalities is $N = \boxed{8}$ .

Let total amount of milk be m and total amount of coffee be c.

Let n be the number of family members.

Since each member drinks 8 L of mixture,

For Ajay,

$\frac{1}{7} m + \frac{2}{17} c = 8$

Also,

$m + c = 8n$

Hence,

$\frac{1}{n} m + \frac{1}{n} c = 8$

Considering the above two equations and Cramer's rule we have,

$m = \frac{\frac{8}{n} - \frac{16}{17}}{\frac{1}{7n} - \frac{2}{17n}}$ , $c = \frac{\frac{8}{7} - \frac{8}{n}}{\frac{1}{7n} - \frac{2}{17n}}$

$m = \frac{\frac{17 \times 8 - 16n}{17 n}}{\frac{3}{7 \times 17 n}}$ , $c = \frac{\frac{8n - 56}{7 n}}{\frac{3}{7 \times 17 n}}$

$m = \frac{7 \times 8 (17 - 2n)}{3}$ , $c = \frac{17 \times 8 (n - 7)}{3}$

For m and c to be positive, $17 - 2n > 0$ and $n - 7 > 0$

$n < \frac{17}{2}$ and $n > 7$

$n < 8.5$ and $n > 7$

Since n is an integer, $\boxed{n = 8}$