Final answer:
To make a 15% bleach solution, 5 gallons of a 5% bleach solution should be added to 10 gallons of a 20% bleach solution.
Step-by-step explanation:
Adding Different Bleach Concentrations
To find out how many gallons of a 5% bleach solution must be added to 10 gallons of a 20% bleach solution to make a 15% bleach solution, we use the concept of mass percentages. The percentage of solute by volume in the final solution must equal the desired concentration. This is a typical dilution problem that can be solved with a simple equation that expresses the balance of bleach in the two solutions before and after they are mixed together.
Let x be the number of gallons of 5% bleach solution. The amount of bleach in the 5% solution is 0.05x gallons, and the amount of bleach in the 20% solution is 10 gallons * 20% = 2 gallons. After mixing, we want a 15% bleach solution, which means the total amount of bleach will be 15% of the combined volume (10 gallons + x gallons).
Setting up the equation looks like this:
0.05x + 2 = 0.15(10 + x)
Solving for x gives us the amount of 5% solution needed:
- Multiply both sides of the equation to get rid of the parentheses: 0.05x + 2 = 1.5 + 0.15x
- Rearrange the equation to solve for x: 0.05x - 0.15x = 1.5 - 2
- Simplify the equation: -0.10x = -0.5
- Divide both sides by -0.10 to find x: x = 5 gallons
Therefore, 5 gallons of a 5% bleach solution are needed to be added to 10 gallons of a 20% bleach solution to get a 15% bleach solution.