Maki uses 20 gallons of a solution with an unknown ethanol concentration and 60 gallons of a [tex]$12 \%$[/tex] ethanol solution to make 80 gallons of a [tex]$10 \%$[/tex] solution. The table shows the amount of each solution.
Ethanol Mixture
| | Gallons | Ethanol Concentration | Total |
| :------ | :------ | :-------------------- | :---- |
| x% Ethanol | 20 | x | |
| 12% Ethanol | 60 | 0.12 | 72 |
| Mixture | 80 | 0.10 | |
What is the value of [tex]$x$[/tex], the concentration of the unknown ethanol solution?
A. 0.02
B. 0.04
Answer (1)
Calculate the amount of ethanol in the 12% solution: 60 × 0.12 = 7.2 gallons.
Calculate the amount of ethanol in the final 10% mixture: 80 × 0.10 = 8 gallons.
Set up the equation: 20 x + 7.2 = 8 , where x is the unknown concentration.
Solve for x : x = 20 0.8 = 0.04 . The concentration of the unknown solution is 0.04 .
Explanation
Problem Analysis We are given a mixture problem where Maki combines 20 gallons of an unknown ethanol concentration with 60 gallons of a 12% ethanol solution to create 80 gallons of a 10% ethanol solution. Our goal is to find the unknown ethanol concentration, denoted as x .
Ethanol in 12% Solution First, calculate the amount of ethanol in the 60 gallons of the 12% solution. This is done by multiplying the volume (60 gallons) by the concentration (0.12): 60 × 0.12 = 7.2 gallons So, there are 7.2 gallons of ethanol in the 12% solution.
Ethanol in Final Mixture Next, calculate the amount of ethanol in the final 80-gallon mixture, which has a 10% concentration. Multiply the total volume (80 gallons) by the concentration (0.10): 80 × 0.10 = 8 gallons Thus, the final mixture contains 8 gallons of ethanol.
Setting up the Equation Now, let's set up an equation to represent the total amount of ethanol in the mixture. The amount of ethanol in the 20 gallons of the unknown solution is 20 x , where x is the ethanol concentration we want to find. The total ethanol in the mixture is the sum of the ethanol from the unknown solution and the 12% solution. Therefore, we have: 20 x + 7.2 = 8
Isolating x Solve the equation for x . First, subtract 7.2 from both sides: 20 x = 8 − 7.2
20 x = 0.8
Solving for x Finally, divide both sides by 20 to find the value of x :
x = 20 0.8 = 0.04
Final Answer Therefore, the concentration of the unknown ethanol solution is 0.04, which means it is a 4% ethanol solution.
Examples
Mixture problems are commonly used in everyday life, such as when mixing different concentrations of cleaning solutions or combining various ingredients in a recipe. For instance, a chef might need to mix a certain amount of a 30% vinegar solution with a 10% vinegar solution to obtain a desired 20% solution for a salad dressing. Understanding how to solve mixture problems allows for precise control over the final product's composition, ensuring the desired outcome is achieved.
