The reaction a+b-> c+d rate=k[a][b]^2 has an initial rate of 0.0810 m/s. what will the initial rate be if [a] is halved and [b] is tripled? what will the initial rate be if [a] is tripled and [b] is halved?

The first to do is to isolate the constant, k, to one side. The rate expression would then be:

k = rate/ [A][B]^2 

This constant, k, will be the same for the given reaction at a certain temperature. Then, we can relate a second rate to this reaction.

rate1/ [A1][B1]^2 = rate2/ [A2][B2]^2 

First question:
0.0810 / [A1][B1]^2 = rate2/ [A2/2][3B2]^2 
rate2 = (0.0810 [A2/2][3B2]^2 )/ [A1][B1]^2
rate2 = 0.3645 M/s

Second Equation:
0.0810 / [A1][B1]^2 = rate2/ [A2/2][3B2]^2 
rate2 = (0.0810 [3A2][B2/2]^2 )/ [A1][B1]^2
rate2 = 0.0608 M/s