To multiply 4 1/6 by 2 1/3, we first need to convert the mixed numbers to improper fractions:

4 1/6 = 25/6
2 1/3 = 7/3

Now, we multiply the two fractions:

(25/6) (7/3) = (257) / (6*3) = 175 / 18

Next, we convert the improper fraction back to a mixed number. The remainder of 175 divided by 18 is 13.

175 ÷ 18 = 9 with a remainder of 13

Therefore, the final result is:

(175 / 18) = 9 13/18