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HomertheGenius
r + R = 10
4/3 r³ π : 4/3 R³ π = 27 : 343
r ³ : R ³ = 27 : 343  / ∛
r : R = 3 : 7
3 R = 7 r
R = 7 r / 3
r + 7 r / 3 = 10   / · 3
3 r + 7 r = 30
10 r = 30
r = 30 : 10
r =  3
Answer: the radius of the smaller sphere is 3.

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