Question

39) The arithmetic mean and geometric mean between two numbers are 75 and 21 respec.
tively, then the numbers are
రెండు సంఖ్యల మధ్య అంక మధ్యమము మరియు గుణాత్మక మధ్యమములు వరుసగా 75 మరియు 21 అయితే ఆ సంఖ్యలు
(2)73,771
3
(1)63,87
అమలం
17,133
(4)3, 147​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{Arithmetic and geometric mean between two}$

$\textsf{numbers are 75 and 21}$

$\underline{\textbf{To find:}}$

$\textsf{The two numbers}$

$\underline{\textbf{Solution:}}$

$\textsf{Let the two numbers be a and b}$

$\mathsf{Arithmetic mean between a and b = 75}$

$\implies\mathsf{\dfrac{a+b}{2}=75}$

$\implies\mathsf{a+b=150}$

$\implies\mathsf{b=150-a}$------(1)

$\textsf{Geometric mean between a and b = 21}$

$\implies\mathsf{\sqrt{ab}=21}$

$\implies\mathsf{ab=441}$-------(2)

$\textsf{Using (1) in (2), we get}$

$\implies\mathsf{a(150-a)=441}$

$\implies\mathsf{150a-a^2=441}$

$\implies\mathsf{150a-a^2-441=0}$

$\implies\mathsf{a^2-150a+441=0}$

$\implies\mathsf{(a-147)(a-3)=0}$

$\implies\mathsf{a=3,147}$

$\mathsf{when a=3,\;b=147}$

$\mathsf{when a=147,\;b=3}$

$\underline{\textbf{Answer:}}$

$\textbf{The required numbers are 3 and 147}$