If 164 is divided in the ratio 32 : 9, what is the value of the smaller part?

The given total amount is $164$, and it is divided into two parts in the ratio $32:9$.
1. Sum of the ratio terms $=32+9=41$ total parts.
2. Value of $1$ ratio unit $=\frac{164}{41}=4$.
3. The smaller part corresponds to the lower ratio term, which is $9$ units.
4. Value of the smaller part $=9\times 4=36$.

Therefore, the smaller part is $36$.