Question

a fuel producer is formulating a fuel mixture that contains 75% ethanol. to make this fuel, they will take 460 gallons of 80% ethanol fuel and add some 55% ethanol fuel. how much of the 55% ethanol fuel should be added? write your answer as a whole number or as a decimal rounded to the nearest tenth. gallons

Explanation:

Step1: Set up the equation

Let \(x\) be the amount of \(55\%\) ethanol fuel. The amount of ethanol in \(80\%\) fuel is \(0.8\times460\), in \(55\%\) fuel is \(0.55x\), and in the final \(75\%\) mixture is \(0.75\times(460 + x)\). So the equation is \(0.8\times460+0.55x=0.75\times(460 + x)\).

Step2: Expand the equation

$$
LATEXBLOCK0
$$

Step3: Solve for \(x\)

Subtract \(0.55x\) from both sides: \(368=345+0.2x\). Then subtract \(345\) from both sides: \(23 = 0.2x\). Divide both sides by \(0.2\): \(x=\frac{23}{0.2}=115\).

Answer:

\(115\)