Ben and frank work in the kitchen of a busy restaurant. ben takes two hours to wash 500 dishes and frank takes three hours to wash 450 dishes. they simultaneously start washing the tremendous stack of 975 dirty dishes from the busy dinner rush. ben leaves work early and frank spends the last half hour of dish washing duty by himself. how long did frank spend washing dishes?

Firstly, we should find the hourly rate of dishes washed for each worker, to do this, I find it useful to use ratios;
For Ben:
500: 2
So 
250: 1
250 dishes per hour

For Frank:
450: 3
So
150: 1
150 dishes per hour

Secondly, We know Frank is going to be doing half an hour of dish washing alone, so during this period, he will wash 75 dishes;
This means when they are both working, they must collectively wash the other 900 dishes;

Now, if we consider the rates of the two workers, we can deduce that they wash 400 dishes per hour working together;
So, to find the time needed to do the 900 they do together, we do:
900/400 = 2.25
They spend 2.25 hr's or 2 hr's 15 min's working together;
We know Frank stays an extra half hour so he works:
2:15 + 0:30 = 2:45
2 hr's 45 min's