SOLUTION: how many gallons of a 15% sugar solution must be mixed with 5 gallons of a 40% sugar solution to make a 30% sugar solution

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Question 384381: how many gallons of a 15% sugar solution must be mixed with 5 gallons of a 40% sugar solution to make a 30% sugar solution
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
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+%28600%2C600%2C-1%2C20%2C-10%2C20%2C.5%2Ax+%2B+6+%2B+%282%2F3%29%29

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.