To determine the ratio in which the grocer must mix the two varieties of rice costing $128$ per kg and $143$ per kg to get a mixture worth $137.75$ per kg, we can use the concept of alligation.

Alligation is a method that enables us to find the ratio in which two or more ingredients at different prices must be mixed to produce a mixture of a desired price.

Let's denote the price of the first variety of rice as $P_1 = 128$ per kg, the price of the second variety as $P_2 = 143$ per kg, and the mean price of the mixture as $P_m = 137.75$ per kg.

The alligation formula is given by:

$\text{Ratio} = \frac{P_2 - P_m}{P_m - P_1}$

Substituting the given values:

$\text{Ratio} = \frac{143 - 137.75}{137.75 - 128}$

Let's calculate the values step by step:

$143 - 137.75 = 5.25$

[ 137.75 - 128