A bourbon that is 51-proof is 25.5% alcohol by volume while one that is 82-proof is 41% alcohol by volume. How many liters of 51 proof bourbon must be mixed with 1.0-L of 82 proof bourbon to produce a 66 proof bourbon? Give answer to three decimal places.

To solve this problem, we'll need to use the concept of dilution.

Let's assume that x liters of 51-proof bourbon are mixed with 1.0-L of 82-proof bourbon.

Step 1: Calculate the total volume of the mixture
The total volume of the mixture will be the sum of the volumes of the two bourbons.
Total volume = x liters + 1.0-L

Step 2: Calculate the total amount of alcohol in the mixture
The total amount of alcohol in the mixture can be found by combining the amount of alcohol in each bourbon.
For the 51-proof bourbon:
Amount of alcohol = 51% of x liters
For the 82-proof bourbon:
Amount of alcohol = 82% of 1.0-L

Step 3: Calculate the alcohol percentage in the mixture
To find the alcohol percentage in the mixture, divide the total amount of alcohol by the total volume of the mixture and multiply by 100.
Alcohol percentage = (Total amount of alcohol / Total volume) * 100

We want the alcohol percentage in the mixture to be 66%.

Now, let's solve the equation:

(51% of x liters + 82% of 1.0-L) / (x liters + 1.0-L) = 66%

Multiply both sides of the equation by (x liters + 1.0-L) to eliminate the denominator:

51% of x liters + 82% of 1.0-L = (x liters + 1.0-L)(66%)

Next, convert the percentages to decimal form:

0.51x + 0.82(1.0-L) = (x + 1.0 - L)(0.66)

Simplify the equation by distributing:

0.51x + 0.82 - 0.82L = 0.66x + 0.66 - 0.66L

Combine like terms:

0.16x + 0.82 - 0.82L = 0.66 - 0.66L

Rearrange the equation:

0.16x - 0.66x = 0.66L - 0.82 + 0.82L - 0.66

Simplify:

-0.50x = 0.16L + 0.16

Divide both sides of the equation by -0.50:

x = (0.16L + 0.16) / -0.50

Simplify further:

x = -0.32L - 0.32 / 0.50

To find the value of x, we need to know the value of L.

To solve this problem, we need to determine the amount of 51-proof bourbon we should mix with 1.0 liter of 82-proof bourbon to obtain a desired concentration of 66 proof.

Let's assume we need "x" liters of 51-proof bourbon to mix with 1.0 liter of 82-proof bourbon.

The proof of a bourbon represents twice the percentage of alcohol by volume. So, we can convert the proof to a decimal to make calculations easier.

For the 51-proof bourbon:
Alcohol by volume = 25.5% = 0.255 (decimal representation)

For the 82-proof bourbon:
Alcohol by volume = 41% = 0.410 (decimal representation)

Now, we can set up an equation based on the alcohol content to solve for "x".

Total alcohol in the mixture = Total alcohol in 51-proof bourbon + Total alcohol in 82-proof bourbon

(0.255 * x) + (0.410 * 1.0) = (0.66 * (1.0 + x))

Let's solve for "x":

0.255x + 0.410 = 0.66 + 0.66x

0.255x - 0.66x = 0.66 - 0.410

-0.405x = -0.25

x = -0.25 / -0.405

x ≈ 0.617

Therefore, you need to mix approximately 0.617 liters of 51-proof bourbon with 1.0 liter of 82-proof bourbon to produce a 66-proof bourbon.