For a big cleaning job, David wants to mix some ammonia-based cleaners to get a certain concentration of ammonia. He has 9 gallons of cleaner that is 20% ammonia. He also has a large supply of cleaner that contains 10% ammonia. How many gallons of the 10%-ammonia cleaner does David need to add to the 20%-ammonia cleaner to obtain a batch that contains 19% ammonia?

Question

Write your answer as a whole number or as a decimal rounded to the nearest tenth.
gallons

Answer 1

To solve this problem, we can set up an equation based on the desired concentration of ammonia in the mixture.

Let \( x \) be the gallons of the 10%-ammonia cleaner that David needs to add.

1. Calculate the amount of ammonia in each cleaner:

   • The 20%-ammonia cleaner has 9 gallons, so the amount of ammonia in it is: \[ 0.20 \times 9 = 1.8 \text{ gallons of ammonia} \]
   • The 10%-ammonia cleaner has \( x \) gallons, so the amount of ammonia in it is: \[ 0.10 \times x = 0.1x \text{ gallons of ammonia} \]

2. Set up the equation for the total amount of ammonia in the final mixture:

   • After mixing, the total volume of the mixture will be \( 9 + x \) gallons.
   • The total amount of ammonia in the mixture will be: \[ 1.8 + 0.1x \text{ gallons of ammonia} \]

3. Set up the equation for the concentration of the final mixture:

   • We want the final concentration to be 19%, which can be expressed as: \[ \frac{1.8 + 0.1x}{9 + x} = 0.19 \]

4. Multiply both sides by \( 9 + x \) to eliminate the fraction: \[ 1.8 + 0.1x = 0.19(9 + x) \]

5. Distribute on the right side: \[ 1.8 + 0.1x = 1.71 + 0.19x \]

6. Rearranging the equation:

   • Move all terms involving \( x \) to one side and constant terms to the other side: \[ 1.8 - 1.71 = 0.19x - 0.1x \] \[ 0.09 = 0.09x \]

7. Solving for \( x \): \[ x = \frac{0.09}{0.09} = 1 \]

David needs to add 1 gallon of the 10%-ammonia cleaner to the 20%-ammonia cleaner to obtain a final mixture that has 19% ammonia.

Thus, the answer is: \[ \boxed{1} \] gallons.