Question

Question 10
The mean of the following distribution is 50.

$\begin{array}{cccccc}x& 10& 30& 50& 70& 90\\ f& 17& 5a+3& 32& 7a-11& 19\end{array}$

Find the value of a and hence the frequencies of 30 and 70.

Answer 1

We know that,
Mean of the distribution $\left(\overline{x}\right)=\frac{{\sum }_{i=1}^5{f}_i{x}_i}{{\sum }_{i=1}^5{f}_i}=50$ [given]
$\Rightarrow \frac{f_1{x}_1+f_2{x}_2+f_3{x}_3+f_4{x}_4+f_5{x}_5}{f_1+f_2+f_3+f_4+f_5}=50$
$\Rightarrow \frac{\left(17\right)\left(10\right)+(5a+3)\left(30\right)+\left(32\right)\left(50\right)+(7a-11)\left(70\right)+\left(19\right)\left(90\right)}{17+5a+3+32+7a-11+19}=50$
$\Rightarrow \frac{170+150a+90+1600+490a-770+1710}{12a+60}=50$
$\Rightarrow \frac{3570-770+640a}{12a+60}=50$
$\Rightarrow 2800+640a=600a+3000$
$\Rightarrow 40a=200\Rightarrow a=5$
$\therefore$ Frequency of 30 $=5a+3=5\left(5\right)+3$ [put a=5]
$=25+3=28$
And frequency of $70=7a-11=7\left(5\right)-11$ [put a=5]
$=35-11=24$
Hence, the frequencies of 30 and 70 are 28 and 24 respectively.