# Julie wants to make 800g of a 15%alcohol solution by mixing a 20% solution and a 40%solution. How many grams of each kind does she need?

Feb 17, 2018

Julie will not be able make a 15% solution using just 20% and 40 solutions to make the mix. Any solution Julie makes using these two components will have an alcohol content of between 20% and 40%.

Feb 17, 2018

Assumption: The target concentration is 20% not 15%

15% material is 640 grams
40% material is 160 grams

#### Explanation:

I am going to show you a sort of cheat way of doing these.

It uses the principles behind a strait line graph and the following concept.

At one end of the blending scale you will have all 15% mix and no 40% mix

At the other end you will have no 15% mix and all 40% mix.

We are interesting in the bit between these.

If you consider just one of the constituents then the other is directly implied as the sum of the two materials is 800g

Using ratio

$\left(\text{change in along")/("change in up}\right) \to \frac{800 - 0}{40 - 15} = \frac{x - 0}{20 - 15}$

$\frac{800}{25} = \frac{x}{5}$

Multiply both sides by 5

$\frac{800}{5} = x = 160$ grams of the 40% material

so the amount of 15% material is $800 - 160 = 640$ grams
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$\textcolor{b r o w n}{\text{Check}}$

$\left(\frac{40}{100} \times 160\right) + \left(\frac{15}{100} \times 640\right)$

$64 + 96 = 160$ pure alcohol from the parts of the blend

$\frac{20}{100} \times 800 = 160$ from the final blend

Both match so correct.