How many liters of glycerin should be added to 12 liters of an $8 \%$ glycerin solution so that the resulting solution contains $16 \%$ glycerin? Round the answer to the nearest hundredth if necessary.
Answer (2)
You need to add approximately 1.14 liters of glycerin to 12 liters of an 8% glycerin solution to achieve a 16% concentration. This was calculated by first determining the initial glycerin content, then setting up a proportion to find out how much more glycerin is required. The final answer rounds to 1.14 liters when calculated accurately.
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Let x be the amount of glycerin to be added.
Set up the equation: 12 + x 0.96 + x = 0.16 .
Solve for x : x = 0.84 0.96 = 7 8 .
Approximate x to the nearest hundredth: x ≈ 1.14 . The final answer is 1.14 .
Explanation
Setting up the problem Let x be the amount of glycerin (in liters) to be added to the 12 liters of 8% glycerin solution. The amount of glycerin in the original solution is 12 × 0.08 = 0.96 liters. When we add x liters of glycerin, the total amount of glycerin becomes 0.96 + x liters. The total volume of the solution becomes 12 + x liters. We want the resulting solution to be 16% glycerin.
Formulating the equation We can set up the equation: 12 + x 0.96 + x = 0.16
Solving for x Now, we solve for x :
0.96 + x = 0.16 ( 12 + x )
0.96 + x = 1.92 + 0.16 x
x − 0.16 x = 1.92 − 0.96
0.84 x = 0.96
x = 0.84 0.96 = 84 96 = 21 24 = 7 8
Finding the approximate value Now, we approximate the value of x to the nearest hundredth:
x = 7 8 ≈ 1.142857
Rounding to the nearest hundredth, we get x ≈ 1.14 liters.
Final Answer Therefore, approximately 1.14 liters of glycerin should be added to the 12 liters of 8% glycerin solution so that the resulting solution contains 16% glycerin.
Examples
Understanding mixture problems is very useful in everyday life. For example, if you are a chef and need to adjust the concentration of a sauce, you can use these principles to calculate how much of an ingredient to add. Similarly, pharmacists use these calculations to prepare medications with specific concentrations. In environmental science, mixture problems can help determine how much pollutant needs to be neutralized to reach a safe level in a water supply. In this case, we calculated that adding approximately 1.14 liters of glycerin to 12 liters of an 8% glycerin solution will result in a 16% glycerin solution. This ensures the final mixture has the desired concentration.
