Find the two numbers whose A.M. is 25 and GM is 20.

#### Solution

\[\text { Let A . M . and G . M . between the two numbers a and b be A and G, respectively } . \]

\[ A = 25\]

\[ \Rightarrow \frac{a + b}{2} = 25\]

\[ \Rightarrow a + b = 50 . . . . . . . (i)\]

\[\text { Also } G = 20\]

\[ \Rightarrow \sqrt{ab} = 20\]

\[ \Rightarrow ab = 400 . . . . . . . (ii)\]

\[\text { Now, putting the value of a in } (ii): \]

\[ \Rightarrow (50 - b)b = 400\]

\[ \Rightarrow b^2 - 50b + 400 = 0\]

\[ \Rightarrow b^2 - 10b - 40b + 400 = 0\]

\[ \Rightarrow b\left( b - 10 \right) - 40\left( b - 10 \right) = 0\]

\[ \Rightarrow \left( b - 10 \right)\left( b - 40 \right) = 0\]

\[ \Rightarrow b = 10, 40\]

\[\text { If }b = 10, \text { then }, a = 400 . \]

\[\text { And , if b = 40, then a } = 10 . \]

\[\text { Thus, the two numbers are 10 and 40 } .\]