Question

A particular guitar string is supposed to vibrate at 210Hz , but it is measured to vibrate at 220Hz

By what percent should the tension in the string be changed to correct the frequency?

Answer 1

v=sqrt(T•L/m)
λ=v/f => f=v/λ= sqrt(T•L/m)/ λ
f₁/f₂={sqrt(T₁•L/m)/ λ}/{sqrt(T₂•L/m)/ λ}
(f₁/f₂)² =T₁/T₂
T₂=T₁(f₂/f₁)²
(T₂-T₁)/T₁=(T₂/T₁) -1=(f₂/f₁)²-1=(220/210)²-1=0.0975.
=> 9.75%

Answer 2

Instead of subtracting 1 at the end, I divided the result by 1 and then subtracted from 1.
1 - (1 / (220/210)²) = 8.89%