Question

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A chef is going to use a mixture of two brands of Italian dressing. The first brand contains 7% vinegar, and the second brand contains 15% vinegar. The chef wants to make 240 milliliters of a dressing that is 13% vinegar. How much of each brand should she use?

Answer 1

Let
`x` be the amount (in mL) of the first brand (7% vinegar) and
`y` the amount of the second brand (15% vinegar). She wants to end up with a mixture with volume 240 mL, so that

`x+y=240`

and she wants it contain 13% vinegar. Each mL of the first brand contributes 0.07 mL (i.e. 7% of 1 mL) vinegar, while each mL of the second brand contributes 0.15 mL (i.e. 15% of 1 mL). The final mixture needs to contribute 0.13 mL (i.e. 13% of 1 mL) for each mL of dressing, so that

`0.07x+0.15y=0.13(x+y)=31.2`

Now solve:

`x+y=240\implies y=240-x`

`0.07x+0.15y=31.2\implies0.07x+0.15(240-x)=31.2`

`\implies-0.08x+36=31.2`

`\implies4.8=0.08x`

`\implies\boxed{x=60}`

`y=240-x\implies\boxed{y=180}`

The chef needs to use 60 mL of the first brand and 180 mL of the second brand.

Answer 2

First brand: 60 milliliters

Second brand: 180 milliliters

Explanation:

Let's call m the amount of the first dressing mark that contains 7% vinegar

Let's call n the amount of the second dressing mark that contains 15% vinegar

The resulting mixture should have 13% vinegar and 240 milliliters.

Then we know that the total amount of mixture will be:

`m + n = 240`

Then the total amount of vinegar in the mixture will be:

`0.07m + 0.15n = 0.13 * 240`

`0.07m + 0.15n = 31.2`

Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -0.15 and add it to the second equation:

`-0.15m -0.15n = 240 * (- 0.15)`

`-0.15m -0.15n = -36`

`-0.15m -0.15n = -36`

+

`0.07m + 0.15n = 31.2`

--------------------------------------

`-0.08m = -4.8`

`m = (-4.8)/(-0.08)`

`m = 60\ milliliters`

We substitute the value of m into one of the two equations and solve for n.

`m + n = 240`

`60 + n = 240`

`n = 180\ milliliters`