The mean of the following data is $50.$ Find the value of a and hence the frequencies of $30$ and $70.$ $\begin{array}{*{20}{c}}X&{10}&{30}&{50}&{70}&{90}\F&{17}&{5a + 3}&{32}&{7a -11}&{19}\end{array}$
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$28$ and $34$
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$68$ and $24$
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$28$ and $24$
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None of these
C
Correct answer
Explanation
Mean = sum(FX)/sum(F) = 50. Sum(F) = 17 + 5a + 3 + 32 + 7a - 11 + 19 = 12a + 60. Sum(FX) = 170 + 30(5a+3) + 1600 + 70(7a-11) + 1710 = 170 + 150a + 90 + 1600 + 490a - 770 + 1710 = 640a + 2800. (640a + 2800)/(12a + 60) = 50 => 640a + 2800 = 600a + 3000 => 40a = 200 => a = 5. Frequencies: 30 is 5(5)+3 = 28, 70 is 7(5)-11 = 24.