One tank is filling at a rate of 3\4 gallon per 2\3 minute. a second tank is filling at a rate of 5/8 gallon per 1/2 minute. which tank is filling faster

To start this, id recommend just calculating each tank at different time intervals to see which is filling faster.  
Let's start at 2 minutes since it's an even number for both scenarios.

Tank A: 3/4 gallon every 2/3 minutes. So for each 2/3 of a minute, we get 3/4 of liquid. So how many 2/3 of a minute are in 2 minutes? The answer is 3 because 2 minutes is the same as 6/3 minutes and 2/3 minutes goes into 6/3 minutes 3 times. So in 2 minutes you get 3 units of the rate. so That's 3/4 + 3/4 +3/4 = 9/4 = 2 1/4 gallons in two minutes.

Tank B: 5/8 gallon per 1/2 a minute. So after 2 minutes we will have 4 units of the rate: 5/8+5/8+5/8+5/8 = 20/8 = 2 4/8= 2 1/2

Since 2 1/2 is more, the second rate is filling faster. 

  

It needs to be changed to common denominators which would be 6/8 x 4/6 which is 6/8 every 40 min and 5/8 every 30 min so they are the same