Find the missing frequency $f_{1}$ and $f_{2}$ in table given below; it is being given that the mean of the given frequency distribution is $145$ .
Class $100 - 120$ $120 - 140$ $140 - 160$ $160 - 180$ $180 - 200$ Total
Frequency $10$ $f_{1}$ $f_{2}$ $15$ $5$ $80$

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Solution

$10 + f_{1} + f_{2} + 15 + 5 = 80$
$f_{1} + f_{2} = 50$ ...(1)
$\frac{(110) (10) + (130) (f_{1}) + (150) (f_{2}) + (170) (15) + (190) (5)}{80} = 145$
$1100 + 130 f_{1} + 150 f_{2} + 2550 + 950 = 11600$
$130 f_{1} + 150 f_{2} = 7000$

$13 f_{1} + 15 f_{2} = 700$
$13 f_{1} + 13 f_{2} = 650$
$-$ $-$ $-$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 2 f_{2} = 50$
$f_{2} = 25$
$f_{1} = 25$

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Class	$100 - 120$	$120 - 140$	$140 - 160$	$160 - 180$	$180 - 200$	Total
Frequency	$10$	$f_{1}$	$f_{2}$	$15$	$5$	$80$