# Suppose a fuel mixture is 5% ethanol and 95% gasoline. How much ethanol (in gallons) must you add to one gallon of the fuel so that the new fuel mixture is 10% ethanol?

Aug 6, 2015

You need to add 0.0556 gallons of ethanol to the mixture.

#### Explanation:

You know that your mixture is 5% ethanol and 95% gasoline.

To determine how much ethanol you need to add to the 1-gallon sample in order to get the percent concetration of ethanol in the mixture to 10%, you first need to determine how much ethanol your sample currently has.

For a 1-gallon sample, you know that

1color(red)(cancel(color(black)("gallon mixture"))) * "5 gallons ethanol"/(100color(red)(cancel(color(black)("gallons mixture")))) = "0.05 gallons ethanol"

Let's say that $x$ represents how muc hethanol you need to add to the mixture. You can express the new percent concentration of ethanol like this

((0.05 + x)color(red)(cancel(color(black)("gallon"))))/((1 + x)color(red)(cancel(color(black)("gallon"))) ) * 100 = 10%

This is equivalent to

$100 \left(0.05 + x\right) = 10 \cdot \left(1 + x\right)$

$0.5 + 10 x = 1 + x$

$9 x = 0.5 \implies x = \frac{0.5}{9} = \textcolor{g r e e n}{\text{0.0556 gallons}}$

So, if you add 0.0556 gallons of ethanol to the 1-gallon sample, you'll get a total mixture of 1.0556 gallons that is 10% ethanol.