# Your chemistry professor gives you a 5 gallon jar containing 2 gallons of 40% alcohol. He asks you to reduce the concentration to 25%. How much water must you add to the jar?

Dec 20, 2015

$3$ gallons

#### Explanation:

The context can be modelled by a general equation:

% "alcohol"="gal alcohol"/"gal solution"

Using the example, 40% "alcohol"=("2 gal alcohol")/("5 gal")*100%, we can set up another equation to solve for the number of galloons added.

Let $x$ be the number of gallons added.

25% "alcohol"=("2 gal alcohol")/("5 gal water"+"x gal water")*100%

25% "alcohol"color(red)(-:100%)=("2 gal alcohol")/("5 gal water"+"x gal water")*100%color(red)(-:100%)

$0.25$ "alcohol"=("2 gal alcohol")/("5 gal water"+"x gal water")

$0.25$ $\text{alcohol"*("5 gal water"+"x gal water")="2 gal alcohol}$

$0.25$ $\text{alcohol} \textcolor{red}{\div 0.25}$color(red)("alcohol")*("5 gal water"+"x gal water")="2 gal alcohol"color(red)(-:0.25)$\textcolor{red}{\text{alcohol}}$

$\text{5 gal water+"x}$ $\text{water}$ $\text{gal"=} 8$ $\text{gal}$

$\text{x gal water}$$=$$\text{3 gal water}$

$\therefore$, $3$ gallons of water must be added.