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Delaney would like to make a 5 lb nut mixture that is 60% peanuts and 40% almonds. She has several pounds of peanuts and several pounds of a mixture that is 20% peanuts and 80% almonds. Let p represent the number of pounds of peanuts needed to make the new mixture, and let m represent the number of pounds of the 80% almond-20% peanut mixture. (a) What is the system that models this situation? (b) Which of the following is a solution to the system: 2 lb peanuts and 3 lb mixture; 2.5 lb peanuts and 2.5 lb mixture; 4 lb peanuts and 1 lb mixture? Show your work.

answer in a very very very very simple answer

3 answers

(a) The system that models the situation is:

  1. \( p + m = 5 \) (The total weight of the mixture is 5 lb)
  2. \( 0.6p + 0.2m = 3 \) (The total pounds of peanuts in the mixture should be 60% of 5 lb, which is 3 lb)

(b) We check each option:

  1. 2 lb peanuts and 3 lb mixture:

    • \( p + m = 2 + 3 = 5 \) (True)
    • \( 0.6(2) + 0.2(3) = 1.2 + 0.6 = 1.8\) (Not true)

  2. 2.5 lb peanuts and 2.5 lb mixture:

    • \( p + m = 2.5 + 2.5 = 5 \) (True)
    • \( 0.6(2.5) + 0.2(2.5) = 1.5 + 0.5 = 2.0\) (Not true)

  3. 4 lb peanuts and 1 lb mixture:

    • \( p + m = 4 + 1 = 5 \) (True)
    • \( 0.6(4) + 0.2(1) = 2.4 + 0.2 = 2.6\) (Not true)

None of the options provided are solutions to the system.

(Score for Question 2: ___ of 5 points) 2. Seiji and Gavin both worked hard over the summer. Together they earned a total of $425. Gavin earned $25 more than Seiji. (a) Write a system of equations for the situation. Use s for the amount Seiji earned and g for the amount Gavin earned. (b) Graph the equations in the system. (c) Use your graph to estimate how much each person earned

answer in a very very very very simple answer

(a) The system of equations is:

  1. \( s + g = 425 \) (Total earnings)
  2. \( g = s + 25 \) (Gavin earned $25 more than Seiji)

(b) The equations can be graphed:

  1. Rearranging the first equation: \( g = 425 - s \)
  2. The second equation is already \( g = s + 25 \)

(c) To estimate:

  • The intersection point of the graphs of the two equations gives us the values for \( s \) and \( g \).

  • From the graph, if you find the point of intersection (where the lines cross), you would estimate:

  • Seiji earned approximately $200,

  • Gavin earned approximately $225.