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The-mean-median-and-mode-of-the-data-values-90-54-x-123-62-78-58-81-are-all-equal-What-is-the-value-of-x-




Question Number 111533 by Aina Samuel Temidayo last updated on 04/Sep/20
The mean,median and mode of the  data values 90,54,x,123,62,78,58,81  are all equal. What is the value of x?
$$\mathrm{The}\:\mathrm{mean},\mathrm{median}\:\mathrm{and}\:\mathrm{mode}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{data}\:\mathrm{values}\:\mathrm{90},\mathrm{54},\mathrm{x},\mathrm{123},\mathrm{62},\mathrm{78},\mathrm{58},\mathrm{81} \\ $$$$\mathrm{are}\:\mathrm{all}\:\mathrm{equal}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}? \\ $$
Commented by Rasheed.Sindhi last updated on 04/Sep/20
78
$$\mathrm{78} \\ $$
Commented by Aina Samuel Temidayo last updated on 04/Sep/20
Your solution please?
$$\mathrm{Your}\:\mathrm{solution}\:\mathrm{please}? \\ $$
Answered by Rasheed.Sindhi last updated on 05/Sep/20
If x has a value other than  90,54,123,62,78,58,81 then the  mode doesn′t exist.  But as the mode exists so  x∈{90,54,123,62,78,58,81}  And also mode is  x(∵ it exist  twice and all others once only)  Now         mean=mode  ((90+54+x+123+62+78+58+81)/8)=x  546+x=8x   x=78  Fortunately it′s also median  123,90,81,78,78,62,58,54  Middle position numbers: 78,78  So,       median=((78+78)/2)=78  Finally,    x= mean=mode=median=78
$${If}\:{x}\:{has}\:{a}\:{value}\:{other}\:{than} \\ $$$$\mathrm{90},\mathrm{54},\mathrm{123},\mathrm{62},\mathrm{78},\mathrm{58},\mathrm{81}\:{then}\:{the} \\ $$$${mode}\:{doesn}'{t}\:{exist}. \\ $$$${But}\:{as}\:{the}\:{mode}\:{exists}\:{so} \\ $$$${x}\in\left\{\mathrm{90},\mathrm{54},\mathrm{123},\mathrm{62},\mathrm{78},\mathrm{58},\mathrm{81}\right\} \\ $$$${And}\:{also}\:{mode}\:{is}\:\:{x}\left(\because\:{it}\:{exist}\right. \\ $$$$\left.{twice}\:{and}\:{all}\:{others}\:{once}\:{only}\right) \\ $$$${Now} \\ $$$$\:\:\:\:\:\:\:{mean}={mode} \\ $$$$\frac{\mathrm{90}+\mathrm{54}+{x}+\mathrm{123}+\mathrm{62}+\mathrm{78}+\mathrm{58}+\mathrm{81}}{\mathrm{8}}={x} \\ $$$$\mathrm{546}+{x}=\mathrm{8}{x} \\ $$$$\:{x}=\mathrm{78} \\ $$$${Fortunately}\:{it}'{s}\:{also}\:{median} \\ $$$$\mathrm{123},\mathrm{90},\mathrm{81},\mathrm{78},\mathrm{78},\mathrm{62},\mathrm{58},\mathrm{54} \\ $$$${Middle}\:{position}\:{numbers}:\:\mathrm{78},\mathrm{78} \\ $$$${So},\:\:\:\:\:\:\:{median}=\frac{\mathrm{78}+\mathrm{78}}{\mathrm{2}}=\mathrm{78} \\ $$$${Finally}, \\ $$$$\:\:{x}=\:{mean}={mode}={median}=\mathrm{78} \\ $$
Commented by Aina Samuel Temidayo last updated on 05/Sep/20
Thanks.
$$\mathrm{Thanks}. \\ $$

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