Question

find rwo numbers such that the mean proportional between them is 12 and the third proportional to them is 96.

Answer 1

Let the required numbers be a and
12 is the mean proportional between a and b
a:12=12:b
ab =144
a = 144/b ....(1)
96 is third proportional to a and b
a:b=b:96
$b ^{2} = 96a \: \: \: \: \: \: \: \: ....(2) \\ put \: the \: value \: of \: "a" \: from \: equ. \: (1) \: in \: equ. \: (2) \\ b^2= 96× \frac{144}{b} \\ b^3= 13824 \\ b =24 \\put \: the \: value \: of \: b \: in \: equ. (1)\\ a= \frac{144}{24} \\ a = 6$