Problem 60 - Intermediate Algebra

Question

Salads. \(\quad A\) chef wants to make 1 gallon ( 128 ounces) of a \(50 \%\) vinegar-to-oil salad dressing. He only has pure vinegar and a mild \(4 \%\) vinegar-to-oil salad dressing on hand. How many ounces of each should he mix to make the desired dressing?

Step-by-Step Solution

Answer

61.33 ounces of pure vinegar and 66.67 ounces of the 4% dressing.

Step 1: Define Variables

Let's define the variables to represent the amounts we need to find. Let \( x \) be the amount in ounces of pure vinegar, and \( y \) be the amount in ounces of the \(4\%\) vinegar-to-oil dressing.

Step 2: Set Up Total Mixture Equation

The chef needs a total of 128 ounces. Setting up an equation for the total amount: \[ x + y = 128 \]

Step 3: Set Up Vinegar Content Equation

We need to account for the vinegar content. The pure vinegar is \(100\%\) or \(1.00\), and the \(4\%\) dressing is \(0.04\). Together, they must equal \(50\%\) of 128 ounces: \[ 1.00x + 0.04y = 0.50 \times 128 \] which simplifies to: \[ x + 0.04y = 64 \]

Step 4: Solve the System of Equations

Now solve the system of equations: \[ x + y = 128 \] \[ x + 0.04y = 64 \] Subtract the second equation from the first: \[(x + y) - (x + 0.04y) = 128 - 64\] Simplifying gives: \[ y - 0.04y = 64 \] \[ 0.96y = 64 \] So, \[ y = \frac{64}{0.96} \approx 66.67 \] Now substitute \( y \approx 66.67 \) back into the equation \( x + y = 128 \) to find \( x \): \[ x + 66.67 = 128 \] \[ x = 128 - 66.67 \approx 61.33 \]

Step 5: Verify the Solution

Ensure the solution meets the conditions. Total vinegar content in \(66.67\) ounces of \(4\%\) dressing is \(0.04 \times 66.67 = 2.67\) ounces. Total vinegar from \(x\) ounces of pure vinegar is \(61.33\) ounces. Total vinegar from both is \(61.33 + 2.67 = 64\) ounces, which is \(50\%\) of 128 ounces. The values satisfy both equations: \(x + y = 128\) and \(x + 0.04y = 64\).

Step 6: Conclusion

The chef should mix approximately 61.33 ounces of pure vinegar and 66.67 ounces of the \(4\%\) vinegar-to-oil dressing to create 1 gallon of the desired dressing.

Key Concepts

Systems of Equations

When faced with situations where you have multiple unknowns and several conditions, systems of equations become incredibly handy. They allow you to solve for two or more variables simultaneously. In our salad dressing problem, we formulated a system of equations to find out how much of each type of dressing the chef should mix. The first equation handles the total amount of liquid, while the second deals with vinegar concentration. Both equations must be satisfied to reach the correct answer.

• The first equation: \( x + y = 128 \) ensures that the total volume is exactly 128 ounces.
• The second equation: \( x + 0.04y = 64 \) ensures the correct vinegar concentration.

When tackling systems of equations, it's important to set up each equation carefully based on the conditions stated in the problem. This method of using more than one equation helps ensure all aspects of the problem are covered and correct solutions are reached.

Mixture Problems

Mixture problems often require a clever setup and understanding of the proportions and total quantities involved. In this scenario, the chef's task is to combine pure vinegar with a mild dressing at a specific concentration to create a desired mixture. To solve such mixture problems, you need to think clearly about each component:

• Each ingredient's contribution to the final mixture must be calculated, particularly when dealing with concentrations.
• The final mixture, in this case, should have specific properties, like a given percentage of vinegar.

Mixture problems like these are frequent in fields like cooking and chemistry, where precise proportions are crucial. They highlight the importance of setting up equations that account for each component's characteristic, whether in terms of quantity or concentration.

Vinegar Concentration

Understanding vinegar concentration in this problem is crucial to reaching the right solution. The concentration tells us how much of a solution is made up of vinegar, expressed here as a percentage or decimal fraction.In our problem:

• Pure vinegar is considered \(100\%\), meaning it's all vinegar with no other substances.
• The mild dressing contains only \(4\%\) vinegar, which means for every 100 ounces, only 4 ounces are vinegar.
• The goal is to create a final solution that is \(50\%\) vinegar.

Controlling and mixing different concentrations to reach a desired level is a common task not only in cooking but also in many other areas, such as chemistry labs and industrial production. It requires precise calculations and an understanding of how each component affects the whole.