Question

kenny, a chemist at a chemical supply company, needs 500 gallons of solution containing 10% acetone. for this purpose, he can mix the two acetone solutions he has on hand. one contains 14% acetone and the other contains 9% acetone. how much of each concentration should he mix together? write your answers as whole numbers or as decimals rounded to the nearest tenth. blank gallons of 14% acetone solution blank gallons of 9% acetone solution

Explanation:

Step1: Set up variables

Let \( x \) be the volume of the \( 14\%\) acetone solution. Then the volume of the \( 9\%\) acetone solution is \( 500 - x\).

Step2: Set up the equation based on the amount of acetone

The amount of acetone in the \( 14\%\) solution is \( 0.14x\), the amount of acetone in the \( 9\%\) solution is \( 0.09(500 - x)\), and the amount of acetone in the final \( 10\%\) solution (which has a volume of \( 500\) gallons) is \( 0.10\times500\). So the equation is \( 0.14x+0.09(500 - x)=0.10\times500\).

Step3: Expand and simplify the equation

Expand \( 0.09(500 - x)\) to get \( 45-0.09x\). The equation becomes \( 0.14x + 45-0.09x=50\). Combine like - terms: \((0.14x-0.09x)+45 = 50\), so \( 0.05x+45 = 50\).

Step4: Solve for \( x\)

Subtract \( 45\) from both sides: \( 0.05x=50 - 45\), so \( 0.05x=5\). Then divide both sides by \( 0.05\): \( x=\frac{5}{0.05}=100\).

Step5: Find the volume of the other solution

If \( x = 100\) (volume of \( 14\%\) solution), then the volume of the \( 9\%\) solution is \( 500 - x=500 - 100 = 400\).

Answer:

100 gallons of \( 14\%\) acetone solution
400 gallons of \( 9\%\) acetone solution