A box contains $10$ pounds of a nut mix that is $50\%$ peanuts, $20\%$ cashews, and $30\%$ almonds. A second nut mix containing $20\%$ peanuts, $40\%$ cashews, and $40\%$ almonds is added to the box resulting in a new nut mix that is $40\%$ peanuts. How many pounds of cashews are now in the box?
$$\textbf{(A) } 3.5 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 4.5 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$$
Difficulty: 9/100 — Straightforward, with some attention to percentages required.
Core Concepts: weighted means, mixtures, system of equations
Challenges: Knowing how to traverse the information given in the problem.
Techniques: Set up a linear equation for the second mixture's weight.
Error-prone Steps: Finding the number of pounds of the second mixture and thinking that is the answer.
Ideal Time:
Experienced: ≤ 20 sec
Intermediate: ≤ 30 sec
Beginner: ≤ 1 min