Question

If the mean proportional between x and y is n times the third proportional, then the ratio x : y will be :

Answer 1

X:Y is n^(2/3).

Mean proportional of X & Y is: sqrt(X*Y).

Third proportional of X & Y is: Y^2 / X

Now sqrt (X * Y) = n * Y^2 /X

X^(1/2) * Y^(1/2) = n * (Y^2)/ X

X^(1/2) * X = n * (Y^2)/ (Y^(1/2))

X^(1/2 + 1) = n * (Y^(2 - 1/2))

X^ (3/2) = n * Y ^(3/2)

(X/Y)^(3/2) = n

((X/Y)^(3/2)) ^ (2/3) = n ^ (2/3)

(X/Y)^ ((3/2)*(2/3))=n^(2/3)

X/Y = n^(2/3)

X:Y ratio is n^(2/3).