Determine one main scale division (msd):

$ 300 \, \text{msd} = 15 \, \text{cm} $

Therefore,

$ 1 \, \text{msd} = \frac{15}{300} \, \text{cm} = 0.05 \, \text{cm} $

Determine one vernier scale division (vsd):

$ 25 \, \text{vsd} = 24 \, \text{msd} $

Therefore,

$ 1 \, \text{vsd} = \frac{24}{25} \, \text{msd} $

Calculate the least count (LC):

The least count is given by the difference between one main scale division and one vernier scale division:

$ \text{LC} = 1 \, \text{msd} - 1 \, \text{vsd} $

Substitute the expression for 1 vsd:

$ \text{LC} = 1 \, \text{msd} - \frac{24}{25} \times 1 \, \text{msd} $

This simplifies to:

$ \text{LC} = \frac{1}{25} \times 1 \, \text{msd} $

Calculate the LC in cm:

Substituting the value of 1 msd:

$ \text{LC} = \frac{1}{25} \times 0.05 \, \text{cm} = 0.002 \, \text{cm} $

Hence, the least count of the traveling microscope is 0.002 cm.