A long solenoid initially has a vacuum core. When a certain...

Let the initial magnetic field inside the solenoid with a vacuum core be $B_0 = \mu_0 n I$.

When the magnetic material is inserted, the magnetic field becomes $B = \mu_0 \mu_r n I = \mu_0 (1 + \chi) n I$.

The fractional increase in the magnetic field is $\dfrac{\Delta B}{B_0} = \dfrac{B - B_0}{B_0} = \chi$.

Given that the percentage increase is $0.15\%$, the fractional increase is $\chi = \dfrac{0.15}{100} = 0.0015$.

The relative permeability of the material is $\mu_r = 1 + \chi = 1 + 0.0015 = 1.0015$.

Answer: $1.0015$