Let's go through the problem step by step to understand which equation Darryl can use to find [tex]\( x \)[/tex], the number of gallons of water he should add to achieve a 5% bleach solution.
1. Understanding the given quantities and goals:
- Darryl starts with 8 gallons of a 10% bleach solution.
- We want to dilute this solution with water until the concentration of bleach is reduced to 5%.
2. Calculate the initial amount of bleach:
- The original solution is 10% bleach, meaning that in 8 gallons, the amount of bleach is:
[tex]\[ \text{Amount of bleach} = 8 \text{ gallons} \times 0.10 = 0.8 \text{ gallons of bleach} \][/tex]
3. Set up the desired final concentration:
- After adding [tex]\( x \)[/tex] gallons of water, the total volume of the solution will be [tex]\( 8 + x \)[/tex] gallons.
- We want this new solution to be 5% bleach. Therefore, the concentration of bleach should be:
[tex]\[ \text{Desired bleach concentration} = 0.05 = \frac{0.8 \text{ gallons}}{8 + x \text{ gallons}} \][/tex]
4. Formulate the equation:
- The equation representing the condition that the new solution is 5% bleach is:
[tex]\[ \frac{0.8}{8 + x} = 0.05 \][/tex]
5. Solve the equation:
- To isolate [tex]\( x \)[/tex], multiply both sides by [tex]\( 8 + x \)[/tex]:
[tex]\[ 0.8 = 0.05 \times (8 + x) \][/tex]
- Distribute the 0.05 on the right side:
[tex]\[ 0.8 = 0.4 + 0.05x \][/tex]
- Subtract 0.4 from both sides to isolate terms involving [tex]\( x \)[/tex]:
[tex]\[ 0.4 = 0.05x \][/tex]
- Divide both sides by 0.05:
[tex]\[ x = \frac{0.4}{0.05} \][/tex]
[tex]\[ x = 8 \][/tex]
Therefore, Darryl should add [tex]\( x = 8 \)[/tex] gallons of water to his solution to achieve the desired 5% bleach concentration.
Hence, the correct equation he can use to determine the amount of water to add is:
[tex]\[ \frac{0.8}{8 + x} = \frac{5}{100} \][/tex]
