Question

an employee at a health food company is making a new type of trail mix. she starts with 10 lbs of hearty mix, which contains 25% dried fruit by weight. she then adds x lbs of active mix, which contains 40% dried fruit by weight. let y be the percent of dried fruit in the new type of trail mix. this function models the situation $y = \frac{2.5 + 0.4x}{10 + x} \cdot 100$. how much active mix should she add in order to have a trail mix containing 30% dried fruit? \boxed{} lbs

Explanation:

Step1: Set up the equation

We know that \( y = 30 \) (since we want 30% dried fruit), so substitute \( y = 30 \) into the formula \( y=\frac{2.5 + 0.4x}{10 + x}\cdot100 \).
We get \( 30=\frac{2.5 + 0.4x}{10 + x}\cdot100 \).

Step2: Simplify the equation

First, divide both sides by 100: \( \frac{30}{100}=\frac{2.5 + 0.4x}{10 + x} \), which simplifies to \( 0.3=\frac{2.5 + 0.4x}{10 + x} \).
Then, multiply both sides by \( 10 + x \): \( 0.3(10 + x)=2.5 + 0.4x \).

Step3: Expand and solve for x

Expand the left - hand side: \( 3+0.3x = 2.5+0.4x \).
Subtract \( 0.3x \) from both sides: \( 3=2.5 + 0.1x \).
Subtract 2.5 from both sides: \( 0.5 = 0.1x \).
Divide both sides by 0.1: \( x=\frac{0.5}{0.1}=5 \).

Answer:

5