How many quarts of a 50% solution of acid must be added to 20 quarts of a 20% solution of acid to obtain a mixture containing a 40% solution of acid? Answer
The Correct Answer and Explanation is:
Let’s solve this problem using the concept of a weighted average.
Given:
- You have 20 quarts of a 20% solution of acid.
- You want to mix it with a 50% solution of acid to obtain a mixture that contains 40% acid.
Let the amount of 50% solution of acid to be added be xxx quarts.
Step 1: Calculate the amount of acid in the two solutions.
- The acid in the 20% solution is: 0.20×20=4 quarts of acid.0.20 \times 20 = 4 \text{ quarts of acid}.0.20×20=4 quarts of acid.
- The acid in the 50% solution is: 0.50×x=0.5x quarts of acid.0.50 \times x = 0.5x \text{ quarts of acid}.0.50×x=0.5x quarts of acid.
Step 2: Calculate the total amount of acid in the mixture.
The total amount of acid after mixing is:4+0.5x quarts of acid.4 + 0.5x \text{ quarts of acid}.4+0.5x quarts of acid.
Step 3: Set up the equation for the desired concentration.
The total volume of the mixture will be 20+x20 + x20+x quarts. We want the mixture to be a 40% solution of acid, so the total amount of acid should be 40% of the total volume. Therefore, we set up the equation:0.40×(20+x)=4+0.5x.0.40 \times (20 + x) = 4 + 0.5x.0.40×(20+x)=4+0.5x.
Step 4: Solve the equation.
Distribute on the left-hand side:0.40×20+0.40×x=4+0.5x,0.40 \times 20 + 0.40 \times x = 4 + 0.5x,0.40×20+0.40×x=4+0.5x,8+0.4x=4+0.5x.8 + 0.4x = 4 + 0.5x.8+0.4x=4+0.5x.
Now, move all terms involving xxx to one side:8−4=0.5x−0.4x,8 – 4 = 0.5x – 0.4x,8−4=0.5x−0.4x,4=0.1x.4 = 0.1x.4=0.1x.
Solve for xxx:x=40.1=40.x = \frac{4}{0.1} = 40.x=0.14=40.
Final Answer:
You need to add 40 quarts of the 50% solution to the 20 quarts of the 20% solution to obtain a mixture containing a 40% solution of acid.
Explanation:
We used a system of equations based on the conservation of mass, where the total amount of acid from both solutions should equal the desired amount of acid in the final mixture. The variable xxx represents the amount of the stronger (50%) acid solution to be added, and the equation was solved to determine the value of xxx.
