Question 384381
There are multiple solutions to this problem because you have 1 equation in 2 unknowns to work with.


Let x = number of gallons of 15% sugar solution.
Let y = number of gallons of 30% sugar solution.


The formula is:


.15*x + .4*5 = .3*y


Simplify to get:


.15*x + 2 = .3*y


Divide both sides of this equation by .3 to get:


.5*x + 6.666666667 = y


This is the same as:


y = .5*x + 6.6666666667


For every value of x, you have a corresponding value of y that will satisfy the equation.


You can graph this equation.


It looks like this:


{{{graph (600,600,-1,20,-10,20,.5*x + 6 + (2/3))}}}


I'll pick 3 points at x = 5, 10, 15 to show you that the relationship holds for all values of x.


When x = 5, y = 9.166666667: .15*x + 2 = 2.75 and .3*y = 2.75. 
When x = 10, y = 11.66666667: .15*x + 2 = 3.5 and .3*y = 3.5.
When x = 15, y = 14.166666667: .15*x + 2 = 4.25 and .3*y = 4.25.


The equation holds for all values of x.


The answer to your question is that:


x gallons of a 15% solution will produce .5*x + 6.66666667 gallons of a 30% solution.


This comes from the equation of y = .5*x + 6.66666667


The x is the number of gallons of 15% solution.
The y is the number of gallons of 30% solution.


If the question had said:


How many gallons of a 15% solution plus 5 gallons of a 40% solution are required to make 15 gallons of a 30% solution, then you could have solved the problem for an exact solution.


In that case, you get y = 15 and the equation of y = .5*x + 6.6666667 becomes:


15 = .5*x + 6.6666667


Now you have 1 equation in one unknown that can be solved.


You subtract 6.66666667 from both sides of the equation to get:


8.33333333 = .5*x


You divide both sides of the equation by .5 to get:


16.66666667 = x


Your answer, in that case, is that 16.66666667 gallons of a 15% solution plus 5 gallons of a 40% solution are required to make 15 gallons of a 30% solution.


.15*16.66666667 = 2.5
.4*5 = 2
2.5 + 2 = 4.5 gallons of solution from the 15% and 40% solution.

.3*15 = 4.5 gallons  of solution from the 30% solution.


Answer is confirmed as correct.