Question

Find two numbers such that the mean proportional between them is $16$ and the third proportional to them is $128$.

Answer 1

Let, the two numbers be $x$ and $y$ respectively

Mean proportional between $x$ and $y$ is $16$

i.e., $x:16=16:y⟹xy=256$

Third proportional to $x$ and $y$ is $128$

i.e., $x:y=y:128⟹y^2=128x$

$⟹y^2=128\times \frac{256}{y}$

$⟹y^3=128\times 256=32768=2^{15}$

$\therefore y=2^5=32$

$⟹x=\frac{256}{y}=\frac{256}{32}=8$

Hence, the two numbers are $8$ and $32$ respectively