# You have 25 milliliters of a drink that contains 70% Caffeine. How many milliliters of a drink containing 40% Caffeine needs to be added in order to have a final drink that is 65% Caffeine?

May 12, 2018

5 ml

#### Explanation:

The key to these dilution problems is to use the idea that the total amount of solute has to be the same before and after mixing. This is the conservation of mass.

Concentration = amount of stuff / volume of solution

$\textsf{c = \frac{m}{v}}$

$\therefore$$\textsf{m = v \times c}$

This means that:

$\textsf{\left(V \times 0.4\right) + \left(25 \times 0.7\right) = 0.65 \times \left(V + 25\right)}$

$\textsf{0.4 V + 17.5 = 0.65 V + 16.25}$

$\textsf{\left(0.65 - 0.4\right) V = 17.5 - 16.25}$

$\textsf{0.25 V = 1.25}$

$\textsf{V = \frac{1.25}{0.25} = 5 \textcolor{w h i t e}{x} m l}$

Iteration check:

Before mixing:

$\textsf{m = \left(25 \times 0.7\right) + \left(5 \times 0.4\right) = 19.5 \textcolor{w h i t e}{x} g}$

After mixing:

$\textsf{m = 30 \times 0.65 = 19.5 \textcolor{w h i t e}{x} g}$

So that's all good.