- Define the system of equations based on the volume balance \(x+y=12\) and the alcohol content balance \(0.8x+0.4y=8.4\).
- Solve the first equation for \(y\) as \(y=12-x\) and substitute this into the second equation.
- Solve the resulting linear equation for \(x\), finding \(x=9\), then determine \(y=3\).
- The required amounts are \(\boxed{9}\) liters of \(80\%\) solution and \(\boxed{3}\) liters of \(40\%\) solution.
Explanation
- Analyzing the Problem and Setting Up the System of Equations
We are given two alcohol solutions with concentrations of \(80\%\) and \(40\%\). We need to mix these to produce a total volume of \(12\) liters of a \(70\%\) alcohol solution. Let \(x\) be the number of liters of the \(80\%\) solution and \(y\) be the number of liters of the \(40\%\) solution.
The total volume of the mixture must be \(12\) liters, giving us the first equation:
\[x + y = 12\]
The amount of pure alcohol from each source must sum to the amount of pure alcohol in the final \(12\)-liter, \(70\%\) solution.
- The amount of pure alcohol in the \(80\%\) solution is \(0.8x\).
- The amount of pure alcohol in the \(40\%\) solution is \(0.4y\).
- The amount of pure alcohol in the final \(12\)-liter solution is \(0.7 \times 12 = 8.4\).
This gives us the second equation:
\[0.8x + 0.4y = 8.4\]
- Solving the System of Equations
We now solve the system of linear equations:
1) \(x + y = 12\)
2) \(0.8x + 0.4y = 8.4\)
From equation (1), we can express \(y\) in terms of \(x\):
\[y = 12 - x\]
Substitute this expression into equation (2):
\[0.8x + 0.4(12 - x) = 8.4\]
Expand and solve for \(x\):
\[0.8x + 4.8 - 0.4x = 8.4\]
\[0.4x + 4.8 = 8.4\]
\[0.4x = 8.4 - 4.8\]
\[0.4x = 3.6\]
\[x = \frac{3.6}{0.4} = 9\]
Now, substitute \(x = 9\) back into the expression for \(y\):
\[y = 12 - 9 = 3\]
- Verification and Final Answer
We found that \(x = 9\) liters and \(y = 3\) liters.
Checking the solution:
- Total volume: \(9 + 3 = 12\) liters. (Correct)
- Total alcohol: \(0.8(9) + 0.4(3) = 7.2 + 1.2 = 8.4\) liters, which is indeed \(70\%\) of \(12\). (Correct)
Thus, you need \(9\) liters of the \(80\%\) alcohol solution and \(3\) liters of the \(40\%\) alcohol solution.
Examples
This problem demonstrates how to calculate the correct proportions of different concentrations of chemicals, which is essential in fields like pharmacy or chemistry when preparing specific mixtures. For instance, if a nurse needs to prepare a specific disinfectant strength or a technician needs to mix reagents for a lab experiment, setting up a system of equations ensures the final mixture meets the exact concentration requirements safely and efficiently. By understanding these linear relationships, one can accurately determine the necessary volumes for any desired mixture.