Eli wants to combine 0.5 gallon of a 10% acid solution with some 35% acid solution to make a 15% acid solution. How much of the [tex]$35 \%$[/tex] acid solution should Eli add?
Answer (2)
Set up the equation: ( 0.10 ) ( 0.5 ) + 0.35 g = 0.15 ( 0.5 + g ) .
Expand the equation: 0.05 + 0.35 g = 0.075 + 0.15 g .
Simplify and isolate g : 0.20 g = 0.025 .
Solve for g : g = 0.20 0.025 = 0.125 . The amount of 35% acid solution Eli should add is 0.125 gallons.
Explanation
Problem Analysis Let's analyze the problem. Eli is mixing two acid solutions of different concentrations to create a new solution with a desired concentration. We need to determine the amount of the 35% acid solution required to achieve the final 15% acid solution.
Define the Variable Let g be the amount (in gallons) of the 35% acid solution that Eli needs to add. The total amount of acid in the final mixture will be the sum of the acid from the 10% solution and the acid from the 35% solution. This total amount must also equal the amount of acid in the final 15% solution.
Set up the Equation We can set up the following equation to represent this situation: ( 0.10 ) ( 0.5 ) + 0.35 g = 0.15 ( 0.5 + g ) Here, ( 0.10 ) ( 0.5 ) represents the amount of acid in the 0.5 gallon of 10% solution, 0.35 g represents the amount of acid in g gallons of 35% solution, and 0.15 ( 0.5 + g ) represents the amount of acid in the final mixture, which is 0.5 + g gallons of 15% solution.
Solve for g Now, let's solve the equation for g :
0.05 + 0.35 g = 0.075 + 0.15 g Subtract 0.15 g from both sides: 0.05 + 0.35 g − 0.15 g = 0.075 + 0.15 g − 0.15 g 0.05 + 0.20 g = 0.075 Subtract 0.05 from both sides: 0.05 − 0.05 + 0.20 g = 0.075 − 0.05 0.20 g = 0.025
Calculate the Value of g Divide both sides by 0.20 to isolate g :
g = 0.20 0.025 g = 0.125 So, Eli needs to add 0.125 gallons of the 35% acid solution.
Final Answer Therefore, Eli should add 0.125 gallons of the 35% acid solution to make the 15% acid solution.
Examples
Mixing solutions of different concentrations is a common task in chemistry, pharmaceuticals, and even cooking. For example, a pharmacist might need to mix different concentrations of a drug to create a specific dosage. Similarly, in cooking, you might mix vinegars of different acidities to achieve a desired flavor profile in a salad dressing. Understanding how to calculate the required amounts ensures the final product has the intended properties.
Eli needs to add 0.125 gallons of the 35% acid solution to create a final solution with a concentration of 15% acid. This was found by setting up an equation based on the acid contents of the solutions. Through calculations, we isolated the variable to find the required amount of the 35% solution.
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