Question

Following data are in increasing order. If their median is 69 , find the value of x.

59,62,65,x,x,+2,72,85,94​

Answer 1

Given Data is: 59, 62, 65, x, x + 2, 72, 85 and 94.

Need to find: The value of x.

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$\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}$⠀

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The formula of Median for even terms is given by :

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$\star\;\boxed{\sf{\pink{Median = \dfrac{\bigg(\dfrac{n}{2} \bigg)^{th} \; observation + \bigg(\dfrac{n}{2} + 1\bigg)^{th} \; observation}{2}}}}$

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• Here, nth term is even that is 8 and Median is 69.

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Now,

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$:\implies\sf Median = \dfrac{\dfrac{\cancel{\;8}}{\cancel{\;2}} \; observation + \bigg(\dfrac{8}{2}^{th} + 1 \bigg)^{th} \; observation}{2} \\\\\\:\implies\sf 69 = \dfrac{4^{th} \; observation + 5^{th}\; observation}{2} \\\\\\:\implies\sf 69 = \dfrac{x + x + 2}{2} \\\\\\:\implies\sf 138 = 2x + 2 \\\\\\:\implies\sf 2x = 138 - 2 \\\\\\:\implies\sf 2x = 136 \\\\\\:\implies\sf x = \cancel\dfrac{136}{2} \\\\\\:\implies{\underline{\boxed{\frak{\pink{x = 68}}}}}\;\bigstar$

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$\therefore{\underline{\sf{Hence, \; the \; required \; value \; of \; x \; is \; \bf{ 68}.}}}$

Answer 2

Given:

• The median of the data 59, 62 ,65 ,x ,x+2 ,72 ,85,94 is 69

To Find:

• the value of x

Solution:

$\pink{\underline{ \mathfrak{As\: We \:know \:that:}}}$

${\longrightarrow} \sf \orange{\: median \: of \:a \: data = \frac{ \frac{n}{2}th \: term + ( \frac{n}{2} + 1)th \: term }{2} \: \bigstar}$

◐ here the variable n is represented by the number of terms in a data.

➺ here, we have 8 terms in the data

◉ so, n = 8

❍ Now, let's substitute the value of 8 in the place of n

${ : \implies} \sf \: median \: = \: \frac{ \frac{n}{2}th \: term + \frac{n}{2 } + 1th \: term }{2} \\ \\ \\ \\ { : \implies} \sf 69 \: = \: \frac{ \frac{8}{2}th \: term + \frac{8}{2} + 1 th \: term }{2} \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \: 69 \: = \frac{4th \: term + 5th \: term}{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

❍ Now, let's substitute the values of the terms

${ \implies} \sf \: 69 = \frac{x + x + 2}{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \: 69 = \frac{2x + 2}{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf 2x + 2 = 69 \times 2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \: 2x + 2 = 138 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \: 2x = 136 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \: x = \cancel \frac{136}{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \: x=\pink{ \underline{ \boxed{68 \bigstar}}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\:$

★ hence the Required value is 68 ★