Your answer is correct but I am not sure I follow your reasoning completely. I think however you have a much better number sense that I would expect from a student in grade 5. Using your line of reasoning I would have said
The number of boats is a multiple of 4 and also a multiple of 7 so it is a multiple of $4 \times 7 = 28.$ Is 28 the answer? One quarter of 28 is 7 and three sevenths of 7 is 3 so this would give 3 red boats. But there are 9 red boats so the number of boats must be 3 times 28 or 84. Check that 84 is correct. One quarter of 84 is 21 and three sevenths of 21 is 9 so 84 is the correct answer.
I think that the teacher expects a "thinking backwards" approach.
After the white boats are removed four sevenths of the remaining are blue so three sevenths are red. Thus three sevenths of the non-white boats is 9 boats so the number of non-white boats is 21. You know that three quarters of the boats are white so one quarter are non-white. Hence the number of boats is 4 times the number of non-white boats, that is $4 \times 21 = 84.$ Again check that 84 is correct.
I hope this helps,
In January 2018 Simoun wrote.
I am quite perplexed with this simple problem and I cannot think of the traditional/basic, one unknown, mathematical equation to come up with "84". Could you please publish the correct mathematical expression for this problem. My 74-year old brains do not function as it used to. Thank you for help.
Many maythematics problems, I think probably most mathematics problems, can't be solved using a "traditional/basic, one unknown, mathematical equation". If equations are used at all in the solution of a problem they commonly involve more then one unknown and there are usually more then one equation involved.
In the solutions Penny gave to this problem you could write some of the steps as equations but the essence of each solution is a logical and mathematical approach to it.