Force between a hollowed lead sphere and a small sphere of mass $m$

I'm having trouble reconciling my solution with other solutions I found online. The problem is from Halliday's Physics book:

The following figure shows a spherical hollow inside a lead sphere of radius $R = 4.00$ cm; the surface of the hollow passes through the center of the sphere before hollowing was $M= 2.95$ kg. With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass $m=0.431$ kg that lies at a distance $d = 9.00$ cm from the center of the lead sphere, on the straight line connecting the centers of the spheres and of the hollow?

MY ATTEMPT: Let's say the hollowed-out lead sphere labelled $1$, while the hollowed out part labelled $2$.

Then $M=m_{1}+m_{2}$. Using density = mass/volume we find that the mass $m_{2}$ is $\frac{M}{8}$.

Then I figured that I could find the force exerted on both objects (hollowed out lead sphere and sphere of mass $m$) as $F = G \frac{m_{1}m}{d^{2}}= G \frac{(M-\frac{M}{8})m}{d^{2}}= G \frac{Mm}{d^{2}} - G\frac{Mm}{8d^{2}}$

OTHER SOLUTIONS: I tried verifying my solution but it seems that other people's solution online is the following $F=G\frac{M m}{d^{2}} - G\frac{\frac{M}{8}m}{(d-R/2)^{2}}$

I get that the $d-R/2$ term would be the distance from the small sphere of mass $m$ to the surface of the lead sphere but wouldn't my original attempt work? Where is the gap in my logic?