# Question 4ede0

Dec 17, 2014

The answer is $3 : 1$ in favor of the 20% solution.

I'll show you two ways of doing this, an easy one and an easier one.

The easiest way of solving this problem is by setting up 2 equations

$a \cdot 20 + b \cdot 60 = 30$ and
$a + b = 1$

SInce we're dealing with mass ratios, we'll assume that the final solution has a fraction a of the 20% solution and a fraction b of the 60% solution; a + b must be equal to one since we're dealing with fractions of a total.

Now, solving the system of equations will produce

$a = 1 - b \to 20 \left(1 - b\right) + 60 b = 30 \to 40 b = 10 \to b = \frac{1}{4}$

And $a = 1 - b = \frac{3}{4}$

So your solution needs $\frac{3}{4}$ of the 20% solution and $\frac{1}{4}$ of the 60% solution; therefore, the mass ratio between the two solutions is $3 : 1$.

The second way of doing this is by using the rule of the cross

You start by placing the starting concentrations on the top row - 20% on top-left, 60% on top-right; you place the desired concentration in the middle - 30%.
Then you substract according to the BLACK lines to get the parts of each solution needed to make the desired one. So,

$30 - 20 = 10$, and $60 - 30 = 30$

The final solution will have $30 + 10 = 40$ parts, out of which

10 parts -> 60% solution - follow the BLUE line on the right;
30 parts -> 20% solution# - follow the BLUE line on the left;

So, once again, the mass ratio between the two solutions will be $3 : 1$ for the 20% solution.