Solution of `9%` acid is to be diluted by adding `3%` acid solution to it. The resulting mixture is to be more than `5%` but less than `7%` acid, If there is `460 L` of the `9%` solution, how many litres of `3%` solution will have to be added

Answer

To solve the problem, we need to determine how many liters of a 3% acid solution must be added to 460 liters of a 9% acid solution so that the resulting mixture has an acid concentration between 5% and 7%.

### Step-by-Step Solution:

1. **Define the Variables**:
Let \( x \) be the amount of 3% acid solution added (in liters).

2. **Calculate the Total Volume of the Mixture**:
The total volume of the mixture after adding \( x \) liters of the 3% solution will be:
\[ 460 + x \text{ liters} \]

3. **Calculate the Amount of Acid in Each Solution**:
- The amount of acid in the 9% solution:
\[ \text{Acid from 9% solution} = 0.09 \times 460 = 41.4 \text{ liters} \]
- The amount of acid in the 3% solution:
\[ \text{Acid from 3% solution} = 0.03 \times x \text{ liters} \]

4. **Set Up the Inequalities for the Acid Concentration**:
The total amount of acid in the mixture will be:
\[ 41.4 + 0.03x \text{ liters} \]
We want the concentration of acid in the mixture to be more than 5% and less than 7%. Therefore, we can set up the following inequalities:
\[ 5\% < \frac{41.4 + 0.03x}{460 + x} < 7\% \]

5. **Convert the Percentages to Equations**:
- For the lower bound (5%):
\[ \frac{41.4 + 0.03x}{460 + x} > 0.05 \]
Multiplying both sides by \( 460 + x \):
\[ 41.4 + 0.03x > 0.05(460 + x) \]
Simplifying:
\[ 41.4 + 0.03x > 23 + 0.05x \]
Rearranging gives:
\[ 41.4 - 23 > 0.05x - 0.03x \]
\[ 18.4 > 0.02x \]
Dividing by 0.02:
\[ x < 920 \]

- For the upper bound (7%):
\[ \frac{41.4 + 0.03x}{460 + x} < 0.07 \]
Multiplying both sides by \( 460 + x \):
\[ 41.4 + 0.03x < 0.07(460 + x) \]
Simplifying:
\[ 41.4 + 0.03x < 32.2 + 0.07x \]
Rearranging gives:
\[ 41.4 - 32.2 < 0.07x - 0.03x \]
\[ 9.2 < 0.04x \]
Dividing by 0.04:
\[ x > 230 \]

6. **Combine the Results**:
From the inequalities, we have:
\[ 230 < x < 920 \]

### Final Answer:
The amount of 3% solution that must be added is between **230 liters and 920 liters**.