Question

a chef is going to use a mixture of two brands of italian dressing. the first brand contains 5% vinegar and the second brand contains 15% vinegar. the chef wants to make 300 milliliters of a dressing that is 9% vinegar. how much of each brand should she use?

Answer 1

The chef will end up with 300 mL of a dressing that is 9% vinegar. Let's calculate how much vinegar will the dressing have:

Therefore,

`v=(300*9)/(100)\rightarrow27`

He'll have 27mL of vinegar.

Now, we know that x mL of the 5% dressing plus y mL of the 15% one will give the chef a total of 27mL of vinegar. Therefore,

`\begin{gathered} x((5)/(100))+y((15)/(100))27 \\ \\ \rightarrow0.05x+0.15y=27 \end{gathered}`

And that x mL of the 5% dressing plus y mL of the 15% one will give the chef a total of 300mL of dressing. Therefore,

`x+y=300`

We have a system of equations:

`\mleft\{\begin{aligned}0.05x+0.15y=27 \\ x+y=300\end{aligned}\mright.`

Let's clear y from equation 2 and replace in equation 1:

`\begin{gathered} x+y=300\rightarrow y=300-x \\ \\ 0.05x+0.15y=27\rightarrow0.05x+0.15(300-x)=27 \\ \rightarrow0.05x+45-0.15x=27\rightarrow-0.10x=-18\rightarrow x=(-18)/(-0.10) \\ \\ \rightarrow x=180 \end{gathered}`

Thereby,

`\begin{gathered} y=300-x\rightarrow y=300-180 \\ \\ \rightarrow y=120 \end{gathered}`

The chef would have to use 180mL of the dressing that's 5% vinegar, and 120mL of the one that's 15% vinegar.