Question

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?

Answer 1

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(0.05 + x) = 10 * (1+x)`

`0.5 + 10x = 1 + x`

`9x = 0.5 => x= 0.5/9 = color(green)("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.