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Volume $$A$$ equals one fourth of the sum of the volumes $$B$$ and $$C$$, while volume $$B$$ equals one sixth of the sum of the volumes $$A$$ and $$C$$. The ratio of volume $$C$$ to the sum of volumes of $$A$$ and $$B$$ is
Let the volumes be $$A,B,C$$. From the two conditions, $$4A=B+C$$ and $$6B=A+C$$. Eliminating $$C$$ gives $$5A=7B$$, and substituting back gives $$23A=7C$$. Taking $$A=7$$ gives $$B=5$$ and $$C=23$$, so $$C\colon(A+B)=23\colon12$$.
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