Question

a chemist has 200 ml of a 10% sucrose solution. she adds x ml of a 40% sucrose solution. the percent - concentration, y, of the final mixture is given by the rational function: $y=\frac{0.1(200)+0.4x}{200 + x}cdot100$. the chemist needs the concentration of the final mixture to be 30%. how many milliliters of the 40% solution should she add to the 10% solution to get this concentration?

Explanation:

Step1: Set up the equation

Set $y = 30$ in the given formula $y=\frac{0.1(200)+0.4x}{200 + x}\cdot100$. So we have $30=\frac{0.1(200)+0.4x}{200 + x}\cdot100$.

Step2: Simplify the equation

First, divide both sides by 100: $\frac{30}{100}=\frac{0.1(200)+0.4x}{200 + x}$, which is $0.3=\frac{20 + 0.4x}{200 + x}$.

Step3: Cross - multiply

$0.3(200 + x)=20 + 0.4x$. Expand the left - hand side: $60+0.3x=20 + 0.4x$.

Step4: Solve for x

Subtract $0.3x$ from both sides: $60=20 + 0.4x-0.3x$. Then $60=20 + 0.1x$. Subtract 20 from both sides: $0.1x=60 - 20=40$. Divide both sides by 0.1: $x = 400$.

Answer:

400