solution is 8%. she has on hand 90ml of a 1% solution and wishes to add 10% solution to obtain the desired 8% solution. how much 10% solution should she add?

To solve this problem, we can use the concept of concentration and volume mixing. Let's set up an equation to find out how much of the 10% solution should be added to the existing 1% solution to achieve an 8% solution.

### Define the Variables
- Let \( x \) be the volume (in ml) of the 10% solution that needs to be added.
- The final solution should be 8%, with the total volume being \( 90 + x \) ml.

### Set Up the Equation
The amount of the active ingredient in each solution is calculated by multiplying the volume by the concentration. We will equate the total amount of the active ingredient from both solutions to the amount in the desired final solution:

1. **Amount of active ingredient in the 1% solution:**  
   \( 90 \times 0.01 = 0.9 \) ml

2. **Amount of active ingredient in the 10% solution:**  
   \( x \times 0.10 = 0.1x \) ml

3. **Amount of active ingredient in the final 8% solution:**  
   \( (90 + x) \times 0.08 \) ml

### Formulate the Equation
Combining all this information, we can set up the equation:

\[
0.9 + 0.1x = 0.08(90 + x)
\]

### Solve the Equation
1. Distribute the \( 0.08 \) on the right side:

\[
0.9 + 0.1x = 7.2 + 0.08x
\]

2. Rearrange the terms to get all \( x \)-terms on one side:

\[
0.1x - 0.08x = 7.2 - 0.9
\]

\[
0.02x = 6.3
\]

3. Solve for \( x \):

\[
x = \frac{6.3}{0.02}
\]

\[
x = 315 \, \text{ml}
\]

### Conclusion
She should add **315 ml** of the 10% solution to the 90 ml of the 1% solution to obtain an 8% solution.

---

Would you like more details on the solution or have any questions?

### Related Questions:

1. What if the desired concentration was 5% instead of 8%?
2. How would the calculation change if she had a different amount of the 1% solution?
3. What is the significance of using the active ingredient concentration in these calculations?
4. How would you check if the final solution indeed has an 8% concentration?
5. Can the equation be set up differently to solve this problem?

**Tip:** When dealing with mixture problems, setting up an equation that balances the amount of the active ingredient helps ensure that the concentrations are mixed correctly.

Learn how to solve a mixture problem involving concentrations: combining 90ml of 1% solution with a calculated amount of 10% solution to achieve an 8% solution. Suitable for high school students learning mixture calculations.

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