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The mean of the following data is $50 .$

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

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

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a=5,

28

and

24

a=5,

28

and

34

None of these

a=3,

68

and

24

answer is C.

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Detailed Solution

Given that the mean is 50 of the data,

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

\Rightarrow

Mean =

\frac{\sum fixi}{\sum fi} = 50

Then,

\Rightarrow 50 = \frac{10 \times 17 + 30 (5 α + 3) + 50 \times 32 + 70 (7 α - 11) + 90 \times 19}{17 + 5 α + 3 + 32 + 7 α - 11 + 19}

\Rightarrow 50 (12 α + 60) = 640 α + 2800

\Rightarrow 3000 + 600 α = 640 α + 2800

\Rightarrow 40 α = 200

\Rightarrow α = 5

frequency of 30,

\Rightarrow 5 α + 3

\Rightarrow 25 + 3

\Rightarrow 28

frequency of

70

\Rightarrow 7 α - 11

\Rightarrow 35 - 11

\Rightarrow 24

Thus, the value of a=5 and the frequencies of 30 and 70 are 28 and 24 respectively.
Hence, the correct option is 3.

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