Sometimes, ratio problems can be solved using not just part to part ratios, but part to whole ratios ( see mathisfun ratios section
https://www.mathsisfun.com/numbers/ratio.html)
In this case, we are given 3/4 pint red to 2/3 pint blue. We add them together to get a whole. Like if we had 3 cats and 2 dogs, the "whole" would be 5 dogs and cats.
So, if we add the 3/4 pint red to 2/4 pint blue we get 17/12 pints of the "whole" mixture.
So, our ratio of part to whole, if we use just red as a part is 3/4 red to 17/12 whole.
We can also think of this as a conversion problem.
We are given 34 pints of a whole mixture (desired result) we can convert that quantity to red pints using our ratio of red parts to whole parts.
We can convert those whole pints to red pints by multiplying by the ratio of red pints to whole pints [ 3/4 pints red/(17/12 whole) ]
the "whole" or purple units cancel leaving the "red" units
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