A chef is going to use a mixture of two brands of Italian dressing. The first brand contains 8% vinegar, and the second brand contains 13% vinegar. The chef wants to make 310 milliliters of a dressing that is 11% vinegar. How much of each brand should she use?

To solve this problem, we need to set up a system of equations. We'll use the method of linear equations to find the amounts of each brand that the chef needs to mix.

Let's denote:
- \( x \) as the amount of the first brand (8% vinegar) to be used.
- \( y \) as the amount of the second brand (13% vinegar) to be used.

We have two pieces of information:
1. The total amount of the mixture should be 310 milliliters.
2. The vinegar concentration in the final mixture should be 11%.

From this, we can set up the following equations:

1. \( x + y = 310 \)
2. \( 0.08x + 0.13y = 0.11 \times 310 \)

Now, let's solve these equations step by step.

### Step 1: Simplify the equations

First, rewrite the second equation to eliminate the percentages:
\[ 0.08x + 0.13y = 34.1 \]

### Step 2: Solve the system of equations

From the first equation:
\[ y = 310 - x \]

Substitute \( y \) in the second equation:
\[ 0.08x + 0.13(310 - x) = 34.1 \]

Simplify and solve for \( x \):
\[ 0.08x + 40.3 - 0.13x = 34.1 \]
\[ 0.08x - 0.13x = 34.1 - 40.3 \]
\[ -0.05x = -6.2 \]
\[ x = \frac{-6.2}{-0.05} \]
\[ x = 124 \]

So, the amount of the first brand (8% vinegar) is 124 milliliters.

Now, substitute \( x \) back into the first equation to find \( y \):
\[ y = 310 - 124 \]
\[ y = 186 \]

So, the amount of the second brand (13% vinegar) is 186 milliliters.

### Conclusion

The chef should use:
- 124 milliliters of the first brand (8% vinegar)
- 186 milliliters of the second brand (13% vinegar)

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:
1. What would be the concentration of vinegar if equal amounts of both brands were used?
2. How would the mixture change if the desired concentration was 10% instead of 11%?
3. What if the chef only had 250 milliliters of the second brand? How would this affect the solution?
4. How can we verify that our solution is correct?
5. What if the total desired volume was different, such as 500 milliliters?
6. How does changing the percentages in each brand affect the mixture?
7. Can we solve this problem graphically, and how?
8. How would the amounts change if there was a third brand with a different concentration?

**Tip:** Always double-check your substituted values in the original equations to ensure the solution is correct.

Learn how to calculate the vinegar concentration in a mixture of two brands of Italian dressing. Solve for the amounts needed of each brand containing different vinegar percentages. Suitable for Grades 9-12.

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