# A chemist needs to create a 25% citrus mixture by adding 20 mL of a 80% citrus solution to a 5% citrus solution. How much of each is needed?

Aug 25, 2015

You would have to use 55 mL of 5 % citrus solution, and we would get 75 mL of 25 % citrus solution.

#### Explanation:

80 % citrus means $\text{80 mL citrus"/"100 mL solution}$

So in 20 mL of 80 % citrus:

$\text{Volume of citrus" = 20 color(red)(cancel(color(black)("mL soln"))) × "80 mL citrus"/(100 color(red)(cancel(color(black)("mL soln")))) = "16 mL citrus}$

In $x \text{ mL}$ of 5 % citrus,

$\text{Volume of citrus" = x color(red)(cancel(color(black)("mL soln"))) × "5 mL citrus"/(100 color(red)(cancel(color(black)("mL soln")))) = x/20 " mL citrus}$

We add $x \text{ mL}$ of 5 % citrus to $\text{20 mL}$ of 80 % citrus to get $20 + x \text{ mL}$ of 25 % citrus.

$\text{Total volume of citrus" = (16 + x/20) " mL}$

$\text{Total volume of solution" = (20+x) " mL}$.

$\frac{16 + \frac{x}{20}}{20 + x} = \frac{25}{100}$

$100 \left(16 + \frac{x}{20}\right) = 25 \left(20 + x\right)$

$1600 + 5 x = 500 + 25 x$

$20 x = 1600 - 500 = 1100$

$x = \frac{1100}{20}$

$x = 55$

We have to add 55 mL of 5 % citrus to 20 mL of 80 % citrus and get 75 mL of 25 % citrus.

Check:

$\frac{16 + \frac{55}{20}}{20 + 55} = \frac{25}{100}$

$\frac{\frac{320 + 55}{20}}{75} = \frac{25}{100}$

$\frac{375}{1500} = \frac{25}{100}$

$\frac{1}{4} = \frac{1}{4}$

It checks!