Question

the literacy rates of 40 cities are given in the following table.If it is given that mean literacy rate is 63.5 then find the missing frequencies x and y

Answer 1

Solution:

$\begin{tabular}{|c|c|c|c|}\cline{1-4}Class&Frequency(fi)&(xi)&xi fi\\ \cline{1-4} 35-40 & 1 & 37.5 & 37.5 \\ \cline{1-4} 40-45 & 2 & 42.5 & 85 \\ \cline{1-4} 45-50 & 3 & 47.5 & 142.5 \\\cline{1-4} 50-55 & x & 52.5 & 52.5x \\\cline{1-4} 55-60 & y & 57.5 & 57.5y \\\cline{1-4} 60-65 & 6 & 62.5 & 375 \\\cline{1-4} 65-70 & 8 & 67.5 & 540 \\\cline{1-4} 70-75 & 4 & 72.5 & 290 \\ \cline{1-4} 75-80 & 2 & 77.5 & 155 \\\cline{1-4} 80-85 & 3 & 82.5 & 247.5 \\\cline{1-4} 85-90 & 2 & 87.5 & 175 \\\cline{1-4} Total & 31+x+y & 1070 & 2047.5+52.5x+57.5y \cline{1-4} \end{tabular}$

Given that Mean = 63.5

$Mean \: = \frac{\Sigma\:x_{i}f_{i}}{\Sigma\:f_{i}} \\ \\ 63.5 = \frac{2047.5 +52.5x + 57.5y }{40} \\ \\ 2540 = 2047.5 + 52.5x + 57.5y \\ \\ 52.5x + 57.5y = 492.5 \\ \\ 10.5x + 11.5y = 98.5 \\ \\ 2.1x + 2.3y = 19.7 \\ \\ 21x + 23y = 197 \: \: \: \: eq1$

total cities are 40

$31 + x + y = 40 \\ \\ x + y = 9 \: \: \: eq2$

From eq1 and eq2

$21x + 23(9 - x) = 197 \\ \\ 21x +207 - 23x = 197 \\ \\ - 2x = - 10 \\ \\x=5$

By this way

$x + y = 9\\ \\ y=9-5\\\\y=4$

Hope it helps you.

Answer 2

Thank you for asking this question. here is your answer:

∑fi = 31 + x + y = 40

x + y = 9

∑fiui = 22 - 2x  - y

Mean = A + ∑fiui / ∑fi x h

= 63.5 = 62.5 + (22-2x-y)/40 x 5

= 2x + y = 14

x = 5

and y = 4

If there is any confusion please leave a comment below.