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Chapter 33: Problem 946

A chemist has an \(18 \%\) solution and a \(45 \%\) solution of a disinfectant. How many ounces of each should be used to make 12 ounces of a \(36 \%\) solution?

Short Answer

Expert verified
The chemist should use 4 ounces of the 18% solution and 8 ounces of the 45% solution to create 12 ounces of a 36% disinfectant solution.

Step by step solution

01

Create Variables

Let's use x and y as the amounts of 18% and 45% solutions, respectively. Our goal is to find the values of x and y that satisfy the following conditions: 1. The total amount of the mixed solution is 12 ounces. 2. The percentage of disinfectant in the mixed solution is 36%.
02

Write the First Equation (Total Amount of Liquid)

We know that the sum of the 18% solution (x) and the 45% solution (y) should equal 12 ounces. Therefore, we can write the first equation as follows: \(x + y = 12\)
03

Write the Second Equation (Percentages of Disinfectant)

To write the second equation, we need to consider the percentages of disinfectant in each solution. We have: - 0.18x: disinfectant in the 18% solution - 0.45y: disinfectant in the 45% solution - 0.36(12): disinfectant in the final 36% solution (12 ounces) We know that the sum of the disinfectant in each initial solution should equal the disinfectant in the final solution. Thus, we can write the second equation: \(0.18x + 0.45y = 0.36(12)\)
04

Solve the System of Equations

Now, we have a system of linear equations: 1. \(x + y = 12\) 2. \(0.18x + 0.45y = 4.32\) To solve this system, we can use the substitution or elimination method. In this case, we will use the substitution method: 1. Solve for x from the first equation: \(x = 12 - y\) 2. Substitute this expression into the second equation: \(0.18(12 - y) + 0.45y = 4.32\) 3. Solve for y: \(2.16 - 0.18y + 0.45y = 4.32\) 4. Combine like terms: \(0.27y = 2.16\) 5. Divide by 0.27: \(y = 8\) Now that we have the value of y, we can find the value of x: 1. Substitute the value of y into the expression for x: \(x = 12 - 8\) 2. Simplify: \(x = 4\)
05

Conclusion

The chemist should use 4 ounces of the 18% solution and 8 ounces of the 45% solution to create 12 ounces of a 36% disinfectant solution.

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