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:
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The total volume of the dressing should be 380 milliliters: \[ x + y = 380 \]
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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:
- \( x + y = 380 \)
- \( 0.08x + 0.13y = 34.2 \)
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.
2 answers