Question

The mean of the following distribution is 50.

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

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

Answer 1

We know that,
Mean = $\frac{{\displaystyle \sum _{}{x}_i{f}_i}}{\sum _{}{f}_i}$

For the following data:

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

Mean = $\frac{\left(10\times 17\right)+\left(30\times \left(5a+3\right)\right)+\left(50\times 32\right)+\left(70\times \left(7a-11\right)\right)+\left(90\times 19\right)}{17+5a+3+32+7a-11+19}$
$$\begin{gathered} \Rightarrow 50=\frac{170+150a+90+1600+490a-770+1710}{60+12a} \\ \Rightarrow 50\left(60+12a\right)=2800+640a \\ \Rightarrow 3000+600a=2800+640a \\ \Rightarrow 640a-600a=3000-2800 \\ \Rightarrow 40a=200 \\ \Rightarrow a=\frac{200}{40} \\ \Rightarrow a=5 \end{gathered}$$

Hence, the value of a is 5.

Also, the frequency of 30 is 28 and the frequency of 70 is 24.