If the numbers 3, 5, 7, 4 have frequencies $x, x + 4, x - 3, x + 8$ with their arithmetic mean of 4, then $x$ will be:

\frac{5}{3}

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\frac{7}{4}

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\frac{2}{3}

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\frac{11}{3}

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Solution

The correct option is A $\frac{5}{3}$
Given:
$\begin{matrix} x_{i} & 3 & 5 & 7 & 4 f_{i} & x & x + 4 & x - 3 & x + 8 \end{matrix}$
$∴ \sum x_{i} f_{i} = 3 x + 5 (x + 4) + 7 (x - 3) + 4 (x + 8) = 3 x + 5 x + 20 + 7 x - 21 + 4 x + 32 = 19 x + 31 \dots (i)$
and $\sum f_{i} = x + x + 4 + x - 3 + x + 8 = 4 x + 9 \dots (i i)$
Given Mean $\frac{\sum f_{i} x_{i}}{\sum f_{i}} = 4$
$\Rightarrow \frac{19 x + 31}{4 x + 9} = 4$ [From Eqs.(i) and (ii)]

$\Rightarrow 19 x + 31 = 16 x + 36 \Rightarrow 3 x = 5 ∴ x = \frac{5}{3}$

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