Gallons Eli wants to combine 0.5 gallon of a 10\% acid solution Acid Amount of with some 35\% acid solution to make a 15\% acid of Concentra- solution. Acid Solution tion How much of the 35\% acid solution should Eli add? 10\% 0.5 0.10 gallon Acid (0.10)(0.5) 35\%
0.35 Acid 0.35
Mixture
0.15 0.15(
+0.5) (0.10)(0.5)+0.35
= 0.15(
+0.5)
The Correct Answer and Explanation is:
To solve this problem, we will use the method of solving by setting up an equation based on the total amount of acid in the mixture.
Problem Restatement
Eli has:
- 0.5 gallon of a 10% acid solution, and
- Wants to add x gallons of a 35% acid solution
- The result should be a 15% acid solution
We are to find the amount of 35% acid solution (x gallons) to add.
Step-by-Step Solution
Let xx be the number of gallons of 35% acid solution that Eli needs to add.
We calculate the total acid content before and after mixing:
- Acid in 10% solution: 0.10×0.5=0.05 gallons of pure acid0.10 \times 0.5 = 0.05 \text{ gallons of pure acid}
- Acid in x gallons of 35% solution: 0.35×x=0.35x gallons of pure acid0.35 \times x = 0.35x \text{ gallons of pure acid}
- Total volume of the new mixture: 0.5+x gallons0.5 + x \text{ gallons}
- Desired concentration in mixture (15%): 0.15×(0.5+x) gallons of pure acid0.15 \times (0.5 + x) \text{ gallons of pure acid}
Now we set up the equation: 0.05+0.35x=0.15(0.5+x)0.05 + 0.35x = 0.15(0.5 + x)
Expand both sides: 0.05+0.35x=0.075+0.15×0.05 + 0.35x = 0.075 + 0.15x
Subtract 0.15×0.15x from both sides: 0.05+0.20x=0.0750.05 + 0.20x = 0.075
Subtract 0.05 from both sides: 0.20x=0.0250.20x = 0.025
Now divide both sides by 0.20: x=0.0250.20=0.125x = \frac{0.025}{0.20} = 0.125
Final Answer:
Eli should add 0.125 gallons (or 1/8 gallon) of the 35% acid solution.
Explanation
This problem is a classic example of mixture problems, where we are combining two solutions of different concentrations to obtain a solution of a desired concentration. The key idea is to track the amount of pure acid in each component and ensure that the total amount of acid in the final mixture matches the desired percentage.
Eli starts with 0.5 gallon of 10% acid solution. Since 10% of this is acid, it contains 0.05 gallons of pure acid. He wants to add some amount xx of 35% solution. The pure acid in this part is 0.35×0.35x. The final mixture will have a total volume of 0.5+x0.5 + x gallons, and we want this to be 15% acid. That means the amount of pure acid in the final mixture must be 0.15(0.5+x)0.15(0.5 + x).
By setting up the equation 0.05+0.35x=0.15(0.5+x)0.05 + 0.35x = 0.15(0.5 + x), we equate the total amount of acid in the mixture to what it must be to reach 15% concentration. Solving the equation step-by-step allows us to isolate xx, the unknown quantity of 35% acid to add.
The result shows that only 0.125 gallons, or 1/8 of a gallon, of the 35% acid solution is needed to achieve the target concentration. This is a relatively small amount because 35% is much stronger than 15%, so not much of it is needed to shift the concentration upward.
This process illustrates how algebra and proportional reasoning are used in real-life applications such as chemistry, pharmacology, and industrial mixing.
