First there are few things to consider; the wording
We raised a 37% solution to a 53% solution, by combining it with a smaller amount of a more concentrated solution. One would expect the concentration of the smaller amount to be higher than that of the final solution.
Since we mixed 6 gals with 9 gals we have a total of 15 gals
We know that our 15 gals is 53% sugar
We know we started with 9 gal at 37% sugar plus 6 gals at x% sugar
So we can just solve for x as follows:
Keeping in mind to write the percents as decimals we have
6x + 0.37(9) = 0.53(15)
6x + 3.33 = 7.95
We can subtract 3.33 from both sides of the equation
6x = 7.95 - 3.33
6x = 4.62
We can dvide both sides by 6
x = 4.62/6
x = 0.77 or 77%
Checking
6(0.77) + 9(0.37) = 15(0.53)
4.62 + 3.33 = 7.95
So 15 gals of 53% sugar solution was made from 6 gals of 77% sugar solution plus 9 gals of 37% sugar solution.
One could also word this as 6 gals of concentrated sugar solution was diluted with 9 gal of 37% sugar solution to make 15 gals of sugar solution at 53%. The process above can be applied to the dilution to find the concentration associated with 6 gals.
I hope this helps if you have any questions please send me a message.
