The transformed equation with integer coefficients whose roots are multiplied by some constant of those of $x^4-\frac{1}{2}{x}^3+\frac{3}{4}{x}^2-\frac{5}{4}x+\frac{1}{16}=0$  is

Required transformed equation is $f\left(\frac{y}{m}\right)=0$

$f\left(\frac{y}{m}\right)={\left(\frac{y}{m}\right)}^4-\frac{1}{2}{\left(\frac{y}{m}\right)}^3+\frac{3}{4}{\left(\frac{y}{m}\right)}^2-\frac{5}{4}\left(\frac{y}{m}\right)+\frac{1}{16}=0$

$=y^4-\frac{1}{2}{y}^{3}m+\frac{3}{4}{y}^2{m}^2-\frac{5}{4}y{m}^3+\frac{1}{16}{m}^4=0$

The exponents of 2 in order are 1,2,2,4

$m=2$

$\therefore y^4-y^3+3y^2-10y+1=0$