How much 90% saline solution should we mix to 3qt. of 15% saline mix to make 45% saline solution?

Sep 17, 2017

$2$ $\text{qt.}$

Explanation:

Let $x$ and $y$ be the number of $\text{qt.}$ of the 90% solution and the number of $\text{qt.}$ of 45% solution, respectively.

We can then express the problem mathematically as:

$R i g h t a r r o w x + 3 = y$

$\mathmr{and}$

$R i g h t a r r o w x$ $\text{qt. of}$ 90% $\text{solution}$ $+ 3$ $\text{qt. of}$ 15% $\text{solution}$ $=$ $y$ $\text{qt. of}$ 45% $\text{solution}$

$R i g h t a r r o w x \times 0.90 + 3 \times 0.15 = y \times 0.45$

$R i g h t a r r o w 0.90 x + 0.45 = 0.45 y$

$R i g h t a r r o w 0.90 x = 0.45 y - 0.45$

Then, let's substitute the first mathematical expression in place of $y$:

$R i g h t a r r o w 0.90 x = 0.45 \left(x + 3\right) - 0.45$

$R i g h t a r r o w 0.90 x = 0.45 x + 1.35 - 0.45$

$R i g h t a r r o w 0.45 x = 0.90$

$\therefore x = 2$

Therefore, we need to mix $2$ $\text{qt.}$ of the 90% saline solution with $3$ $\text{qt.}$ of 15% saline solution to make a 45% solution.

Sep 17, 2017

We need $2$ qt. of 90% saline solution.

Explanation:

Let us assume that we need to mix $x$ qt. of 90% saline solution. It will have $x \times \frac{90}{100} = 0.9 x$ qt. saline.

It is mixed with $3$qt. of 15% saline solution, this as $3 \times \frac{15}{100} = 0.45$ qt. saline

and on mixing the two, the total quantum of saline is $0.9 x + 0.45$ qt.

This is 45% of $x + 3$ or $\left(x + 3\right) \times 0.45$

hence $0.45 \left(x + 3\right) = 0.9 x + 0.45$

or $0.45 x + 1.35 = 0.9 x + 0.45$

or $0.9 x - 0.45 x = 1.35 - 0.45$

or $0.45 x = 0.9$

i.e. $x = \frac{0.9}{0.45} = 2$

Hence, we need $2$ qt. of 90% saline solution.