### Mean, Median Range with one Unknown

Question Sample Titled 'Mean, Median Range with one Unknown'

${\left\lbrace{a}-{8},{a}-{2},{a},{a}+{3},{a}+{4},{a}+{4}\right\rbrace}$ and ${\left\lbrace{a}-{9},{a}-{4},{a}-{1},{a}+{2},{a}+{5},{a}+{8}\right\rbrace}$ are two groups of numbers. Which of the following is/are true?

 I. The two groups of numbers have the same mean. II. The two groups of numbers have the same median. III. The two groups of numbers have the same range.

A
I only
B
None of the above
C
I, II and III
D
I and III only

 I. The mean of first group of numbers  $={\left(\dfrac{{{6}{a}+{1}}}{{6}}\right)}$  $={a}+\dfrac{{1}}{{6}}$ The mean of the second group of numbers  $={\left(\dfrac{{{6}{a}+{1}}}{{6}}\right)}$  $={a}+\dfrac{{1}}{{6}}$ ∴   Their means are equal. II. Median of the first group of numbers  $=\dfrac{{{a}+{a}+{3}}}{{2}}$  $={a}+{1.5}$ Median of the second group of numbers  $=\dfrac{{{a}-{1}+{a}+{2}}}{{2}}$  $={a}+{0.5}$ ∴   Their medians are not equal. III. Range of the first group of numbers  $={4}+{8}$  $={12}$ Range of the second group of numbers  $={8}+{9}$  $={17}$ ∴   Their ranges are not equal.

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