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Question Number 9577 by j.masanja06@gmail.com last updated on 18/Dec/16

The mean of n number is 20.if   the same numbers together with 30  give a new mean of 22,  find n.

$$\mathrm{The}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{n}\:\mathrm{number}\:\mathrm{is}\:\mathrm{20}.\mathrm{if}\: \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{numbers}\:\mathrm{together}\:\mathrm{with}\:\mathrm{30} \\ $$$$\mathrm{give}\:\mathrm{a}\:\mathrm{new}\:\mathrm{mean}\:\mathrm{of}\:\mathrm{22}, \\ $$$$\mathrm{find}\:\mathrm{n}. \\ $$

Answered by ridwan balatif last updated on 18/Dec/16

first condition  x_n ^− =20  ((x_1 +x_2 +...+x_n )/n)=20  x_1 +x_2 +...+x_n =20n...(∗)  second condition  x_(n+1) ^− =22  (((x_1 +x_2 +...+x_n )+30)/(n+1))=22  (20n)+30=22n+22  2n=8  n=4

$$\mathrm{first}\:\mathrm{condition} \\ $$$$\overset{−} {\mathrm{x}}_{\mathrm{n}} =\mathrm{20} \\ $$$$\frac{\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +...+\mathrm{x}_{\mathrm{n}} }{\mathrm{n}}=\mathrm{20} \\ $$$$\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +...+\mathrm{x}_{\mathrm{n}} =\mathrm{20n}...\left(\ast\right) \\ $$$$\mathrm{second}\:\mathrm{condition} \\ $$$$\overset{−} {\mathrm{x}}_{\mathrm{n}+\mathrm{1}} =\mathrm{22} \\ $$$$\frac{\left(\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +...+\mathrm{x}_{\mathrm{n}} \right)+\mathrm{30}}{\mathrm{n}+\mathrm{1}}=\mathrm{22} \\ $$$$\left(\mathrm{20n}\right)+\mathrm{30}=\mathrm{22n}+\mathrm{22} \\ $$$$\mathrm{2n}=\mathrm{8} \\ $$$$\mathrm{n}=\mathrm{4} \\ $$

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