The mean of the following distribution is $50$. Find the value of $a$ and hence the frequencies of $30$ and $70$.

$x$	$f$
$10$	$17$
$30$	$5a+3$
$50$	$32$
$70$	$7a-11$
$90$	$19$

(A) To find the mean,we use the formula $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.
Constructing the frequency table:

$x_i$	$f_i$	$f_i x_i$
$10$	$17$	$170$
$30$	$5a+3$	$150a+90$
$50$	$32$	$1600$
$70$	$7a-11$	$490a-770$
$90$	$19$	$1710$
Total	$\sum f_i = 12a + 60$	$\sum f_i x_i = 640a + 2710$

Given $\bar{x} = 50$,we have:
$50 = \frac{640a + 2710}{12a + 60}$
$50(12a + 60) = 640a + 2710$
$600a + 3000 = 640a + 2710$
$3000 - 2710 = 640a - 600a$
$290 = 40a$
$a = \frac{290}{40} = 7.25$
Frequency of $30 = 5(7.25) + 3 = 36.25 + 3 = 39.25$
Frequency of $70 = 7(7.25) - 11 = 50.75 - 11 = 39.75$