Question

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 380 milliliters of a dressing that is 9% vinegar. How much of each brand should she use?

Answer 1

To solve the problem, we will use a system of equations based on the information given about the two brands of Italian dressing.

Let \( x \) be the amount (in milliliters) of the first brand (8% vinegar) that the chef will use, and \( y \) be the amount (in milliliters) of the second brand (13% vinegar).

We have two main conditions based on the problem:

1. The total volume of the dressing should be 380 milliliters:
\[
x + y = 380
\]

2. The total amount of vinegar in the mixture should equal 9% of 380 milliliters. The amount of vinegar from the first brand is \( 0.08x \), and from the second brand, it is \( 0.13y \):
\[
0.08x + 0.13y = 0.09 \times 380
\]

Calculating \( 0.09 \times 380 \):
\[
0.09 \times 380 = 34.2
\]
So, the second equation becomes:
\[
0.08x + 0.13y = 34.2
\]

Now we have the following system of equations:

1. \( x + y = 380 \)
2. \( 0.08x + 0.13y = 34.2 \)

We can solve this system using substitution or elimination. Let's solve for \( y \) from the first equation:

\[
y = 380 - x
\]

Now, substitute \( y \) into the second equation:

\[
0.08x + 0.13(380 - x) = 34.2
\]

Distributing the \( 0.13 \):

\[
0.08x + 49.4 - 0.13x = 34.2
\]

Combining like terms:

\[
-0.05x + 49.4 = 34.2
\]

Subtracting 49.4 from both sides:

\[
-0.05x = 34.2 - 49.4
\]
\[
-0.05x = -15.2
\]

Dividing by -0.05:

\[
x = \frac{-15.2}{-0.05} = 304
\]

Now that we have \( x \), we can find \( y \):

\[
y = 380 - x = 380 - 304 = 76
\]

Thus, the chef should use:
- **304 milliliters** of the first brand (8% vinegar),
- **76 milliliters** of the second brand (13% vinegar).

To verify, we can check the total amount of vinegar:

For the first brand:
\[
0.08 \times 304 = 24.32 \text{ milliliters}
\]

For the second brand:
\[
0.13 \times 76 = 9.88 \text{ milliliters}
\]

Adding them together:
\[
24.32 + 9.88 = 34.2 \text{ milliliters}
\]

This is indeed 9% of 380 milliliters. Our calculations are correct.

So the final answer is:
- **304 milliliters** of the first brand,
- **76 milliliters** of the second brand.