Question

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?
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.