clearly define variables. set up the equations.How many gallons of a 30% alcohol solution must be mixed with 60 gallons of a 14% solution to obtain a solution that is 20% alcohol?
x represents?
y represents?
equation 1?
equation 2?
amount of 30% solution --- x gal
amount of 14% stuff ------ 60-x
solve for x
.30x + .14(60-x) = .20(60)
or if you want two variables:
x+y = 60
.30x + .14y = .20(60)
To solve this problem, let's define the variables and set up the equations.
x represents the number of gallons of the 30% alcohol solution that needs to be added.
y represents the final volume of the alcohol solution mixture, in gallons.
Now, let's set up the equations:
Equation 1: This equation represents the total amount of alcohol in the mixture.
The alcohol in the 30% solution (x gallons) + the alcohol in the 14% solution (60 gallons) = the alcohol in the final mixture (y gallons).
0.30x + 0.14(60) = 0.20y
Explanation:
- 0.30x is the amount of alcohol in the 30% solution (since the solution is 30% alcohol).
- 0.14(60) is the amount of alcohol in the 14% solution (since the solution is 14% alcohol and the total volume is 60 gallons).
- 0.20y is the amount of alcohol in the final mixture (since we want the final solution to be 20% alcohol).
Equation 2: This equation represents the total volume of the mixture.
The volume of the 30% solution (x gallons) + the volume of the 14% solution (60 gallons) = the final volume of the mixture (y gallons).
x + 60 = y
Explanation:
- x is the volume of the 30% solution we want to add.
- 60 is the volume of the 14% solution we already have.
- y is the final volume of the mixture, which is the sum of the two volumes.
So, the equations are:
0.30x + 0.14(60) = 0.20y
x + 60 = y
These equations represent the alcohol content and the volume of the mixture in terms of x and y, allowing us to solve for both variables.