Powder Size and Fineness Relationship

Powder fineness is inversely proportional to its particle size. This means that as a powder becomes finer, its individual particles become smaller.

Analyzing Powder Sizes

The question establishes a baseline:

• A coarse powder is assigned a relative size of $1$.

We need to find the size corresponding to a very fine powder among the given options. A very fine powder will have a particle size significantly smaller than the coarse powder.

Comparing Particle Size Options

The options provided are fractions, representing sizes relative to the coarse powder:

• Option 1: $\frac{1}{24}$
• Option 2: $\frac{1}{200}$
• Option 3: $\frac{1}{6}$
• Option 4: $\frac{1}{90}$

To determine the finest powder, we must identify the smallest fraction. Comparing the fractions:

• $\frac{1}{200}$ is smaller than $\frac{1}{90}$
• $\frac{1}{90}$ is smaller than $\frac{1}{24}$
• $\frac{1}{24}$ is smaller than $\frac{1}{6}$

Therefore, $\frac{1}{200}$ represents the smallest particle size among the options.

Conclusion on Powder Size

Since a very fine powder corresponds to the smallest particle size, the size for a very fine powder is $\frac{1}{200}$.