Problem 68 - Precalculus Mathematics for Calculus

Question

A biologist has two brine solutions, one containing \(5 \%\) salt and another containing \(20 \%\) salt. How many milliliters of each solution should she mix to obtain \(1 \mathrm{L}\) of a solution that contains \(14 \%\) salt?

Step-by-Step Solution

Answer

The biologist should mix 400 mL of the 5% solution with 600 mL of the 20% solution.

Step 1: Define Variables

Let \(x\) represent the milliliters of the 5% salt solution, and \(y\) represent the milliliters of the 20% salt solution.

Step 2: Set Up Equation 1

The total volume of the solution should be \(1000\) milliliters. Hence, the equation is \(x + y = 1000\).

Step 3: Set Up Equation 2

The desired salt concentration is 14%. Thus, the equation for salt content is \(0.05x + 0.20y = 0.14 \times 1000\). This simplifies to \(0.05x + 0.20y = 140\).

Step 4: Solve System of Equations

We have the system of equations: \(x + y = 1000\) and \(0.05x + 0.20y = 140\). Solve the first equation for \(y\): \(y = 1000 - x\). Substitute \(y\) in the second equation: \(0.05x + 0.20(1000 - x) = 140\).

Step 5: Simplify and Solve for \(x\)

Expand and simplify: \(0.05x + 200 - 0.20x = 140\), which simplifies to \(-0.15x + 200 = 140\). Solve for \(x\): \(-0.15x = -60\), so \(x = 400\).

Step 6: Solve for \(y\)

Substitute \(x = 400\) into \(y = 1000 - x\) to find \(y\): \(y = 1000 - 400 = 600\).

Step 7: Verify Solution

Check that the salt content equation holds: \(0.05 imes 400 + 0.20 imes 600 = 20 + 120 = 140\), which matches the desired 14% of 1000 mL (140 mL). Thus, the solution is verified.

Key Concepts

System of Equations

Mixture problems often require solving systems of equations. This typically involves more than one variable and equation. In our problem, we deal with two unknowns: \(x\) and \(y\), representing the milliliters of 5% and 20% salt solutions, respectively.
To form our equations, we first consider the total volume:

• The sum of the volumes of the two solutions must equal 1000 milliliters, giving us our first equation: \(x + y = 1000\).

Next, we use the information about the concentration:

• We want the final salt concentration to be 14% of the total 1 liter. This is reflected in the equation \(0.05x + 0.20y = 140\), representing the total amount of salt in the solutions combined.

Solving these equations involves substituting one equation into another, allowing you to find the values of \(x\) and \(y\) that satisfy both conditions.

Salt Concentration

Understanding salt concentration is crucial for solving this type of problem. Concentration is usually given as a percentage and indicates the ratio of the solute (salt) to the solution. In this scenario:

• The 5% solution means that each milliliter contains 0.05 grams of salt.
• Similarly, the 20% solution means each milliliter contains 0.20 grams of salt.

For mixture problems, you'll often need to create an equation reflecting the total desired concentration.

• Our target is a 14% concentration in 1 liter (1000 milliliters). Thus, the equation \(0.05x + 0.20y = 140\) aligns with achieving 140 grams of salt in total.

This equation ensures that when you combine the two solutions, the salt quantity matches the desired concentration in the new solution.

Solution Verification

Once you have a solution, verifying it ensures your understanding and accuracy. Verification can be as simple as rechecking the math and making sure everything aligns logically.
With our solutions, \(x = 400\) mL and \(y = 600\) mL, we need to confirm:

• Total volume: Substitute \(x = 400\) and \(y = 600\) back into the equation \(x + y = 1000\). This checks out since 400 + 600 = 1000 mL.
• Salt content: Check if the salt equation \(0.05x + 0.20y = 140\) holds true. Calculating gives \(0.05 \times 400 + 0.20 \times 600 = 20 + 120 = 140\) grams of salt, which is exactly 14% of 1 liter.

Thus, the calculated values fulfill both the total volume and the desired concentration, confirming our results are correct.