Suppose that you have a supply of a 20% solution of alcohol and a 80% solution of alcohol. How many quarts of each should be mixed to produce 60 quarts that is 65% alcohol?
amount of 20% stuff ---- x qrts
amount of 80% stuff = 60-x qrts
solve for x
.2x + .8(60-x) = .65(60)
To solve this problem, we need to set up an equation based on the amount of alcohol in the two solutions.
Let's assume that you will mix x quarts of the 20% solution and y quarts of the 80% solution to produce 60 quarts of a 65% solution.
First, let's determine the total amount of alcohol in the final mixture. We know that 60 quarts of the final mixture contain 65% alcohol, so the amount of alcohol is:
0.65 * 60 = 39 quarts
Next, let's establish equations based on the amount of alcohol in each solution.
In the 20% solution, the amount of alcohol is given by 0.20 * x.
In the 80% solution, the amount of alcohol is given by 0.80 * y.
Since we want a total of 39 quarts of alcohol, we can write the following equation:
0.20x + 0.80y = 39
We also know that the total volume of the mixture is 60 quarts, so we can write a second equation:
x + y = 60
Now, we have a system of two equations:
0.20x + 0.80y = 39 ---(equation 1)
x + y = 60 ---(equation 2)
To solve this system of equations, we can use the method of substitution or elimination.
Let's solve it using the method of substitution:
From equation 2, we have y = 60 - x.
Substituting this value of y into equation 1, we get:
0.20x + 0.80(60 - x) = 39
Simplifying:
0.20x + 48 - 0.80x = 39
Combine like terms:
0.60x + 48 = 39
Subtract 48 from both sides:
0.60x = 39 - 48
0.60x = -9
Divide both sides by 0.60:
x = -9 / 0.60
x = -15
Since we cannot have a negative number of quarts, the value of x is not valid.
Thus, there is no solution to this particular problem.
In conclusion, it is not possible to mix a 20% solution and an 80% solution of alcohol to produce a 60-quart mixture that is 65% alcohol.
To solve this problem, let's assume you need to mix x quarts of the 20% solution and y quarts of the 80% solution to obtain 60 quarts of a 65% alcohol solution.
Step 1: Write down the given information and the unknowns:
x + y = 60 (The sum of the quantities of the two solutions equals the total quantity required)
0.20x + 0.80y = 0.65 * 60 (The sum of the quantities of alcohol in the two solutions is equal to 65% of the total quantity required)
Step 2: Simplify the equations:
x + y = 60
0.20x + 0.80y = 39
Step 3: Solve the first equation for x:
x = 60 - y
Step 4: Substitute the value of x in the second equation:
0.20(60 - y) + 0.80y = 39
12 - 0.20y + 0.80y = 39
0.60y = 27
y = 27 / 0.60
y = 45
Step 5: Substitute the value of y in the first equation to find x:
x + 45 = 60
x = 15
Step 6: Check your solution:
0.20 * 15 + 0.80 * 45 = 3 + 36 = 39, which is equal to 0.65 * 60.
Therefore, you should mix 15 quarts of the 20% solution and 45 quarts of the 80% solution to obtain 60 quarts of a 65% alcohol solution.