# Mom made a 5% bleach solution with 50 ml of bleach and 950 ml water. How much bleach should be added to this in order to obtain a 15% bleach solution?

Jun 6, 2018

$117.6$ ml

#### Explanation:

The current solution has $50$ ml of bleach and a total volume of $50 + 950 = 1000$ ml. If we add a certain quantity $b$ of bleach, we will have more bleach, but also more solution in general: the ratio bleach-to-total volume will change as follows:

$\setminus \frac{50 + b}{1000 + b}$

we want this ratio to equal 15% = 0.15, so we must ask

$\setminus \frac{50 + b}{1000 + b} = 0.15$

Solve for $b$: if we multiply both sides by $1000 + b$ we have

$50 + b = 0.15 \left(1000 + b\right) = 150 + 0.15 b$

Subtract $0.15 b$ from both sides:

$50 + b - 0.15 b = 150$

Subtract $50$ from both sides:

$b - 0.15 b = 150 - 50$

Simplify both sides:

$0.85 b = 100$

Divide both sides by $0.85$

$b = \setminus \frac{100}{0.85} \setminus \approx 117.6$

Let's check our answer: the new solution will have $50 + 117.6 = 167.6$ ml of bleach, for a total volume of $1000 + 117.6 = 1117.6$ ml. The ratio is

$\setminus \frac{167.6}{1117.6} \setminus \approx 0.15$

so we're good!