After Jack added 15.2 gallons of water to a half-full bucket of water, the total volume of water in the bucket was 3734 gallons.
I think you are missing a question here. If I were to guess, the question is "What's the total volume of the bucket?" Right? I will assume that is the question.
As with all word problems, there are two parts. The first part is to express the words, phrases and sentences in terms of mathematical (algebraic equations, or equations with variables/unknowns), and then to solve to the unknowns (the variables).
In the one sentence, you have the first dependent ("after...") clause, and then the second independent ("...the total volume") clause.
From the first part, the bucket is half-full...that is, it is already at half its total volume!! If the bucket were full, it would have water at its total volume!
So the words half-full, full, and so on, relate to the volume of a container.
Let's now say that the total volume = the variable V.
We also know that if add ("...Jack added...") 15.2 gallons to half the volume, the resulting volume will be 3734 gallons.
We can represent the half-full nature of the bucket as the product 0.5 * V, or 0.5V.
If we add 15.2 gallons to it, then this becomes the mathematical expression: 0.5V + 15.2,
For now, I omit the dimensions or units "gallons" to keep the expression uncomplicated.
The resulting volume is 3734 gallons. When you get a result in a word problem, this usually implies and equal sign....the creation of an equation!
So the expression becomes the equation:
0.5V + 15.2 = 3734
If you were asked the total volume of the bucket V, then you have all the information in that one equation now to get the answer. It is a rule that you need n equations for n unknowns: you have one unknown (the variable V), and you have one equation.
The rest is algebra.
* Subtract 15.2 from both sides of the equation:
A. 0.5V + 15.2 - 15.2 = 3734 - 15.2
B. 0.5V = 3718.8
* Now to get V (or 1 * V), divide both sides of the equation by 0.5
C. 0.5V / 0.5 = 3718.8 / 0.5
D. V = 7437.6
If you were asked for the total container/bucket volume, as I suspect, then your answer is 7437.6 gallons.