Question

The A.M. of two numbers exceeds their G.M. by $15$ and H.M. by $27$, find the bigger of the two numbers.

Answer 1

Let the numbers be a, b and their A.M., G.M., and H.M. be denoted by
A,G, and H respectively. Also we know that A.G.H are in G.P.
or $G^2=AH$ ...(1)
Since $A-G=15$ and $A-H=27$
${(A-15)}^2=G^2=AH$ by (1) =A(A-27)
or $-30A+225=-27Aor3A=225$
$\therefore A=75=\frac{a+b}{2}\therefore a+b=150$ ...(2)
Since $A-G=15\therefore 75-G=15$
or $G=60=\sqrt{ab}$
$ab=3600...\left(3\right)$
Hence from (2) and (3) we conclude that a and b are the roots of $t^2-150t+3600=0$
or $(t-120)(t-30)=0\therefore t=120,30$
Hence the two numbers are $120$ and $30$.