Real-World Systems of Inequalities: Contextual Solutions

A farmer has 100 acres of land to plant corn and wheat. Each acre of corn requires 2 hours of labor, and each acre of wheat requires 3 hours of labor. If the farmer has a total of 240 hours of labor available, what are the possible combinations of corn and wheat he can plant?

• (10, 80)

• (20, 70)

• (30, 60)

• Possible combinations of corn and wheat are: (0, 80), (10, 70), (20, 60), (30, 50), (40, 40), (50, 30), (60, 20), (70, 10), (80, 0) where (x, y) represents (corn acres, wheat acres).

A school is organizing a field trip and has a budget of $500. The cost per student is $20 for transportation and $15 for admission. If the school wants to take at least 20 students, what is the maximum number of students they can take on the trip?

• 15

• 0

• 25

• 30

A gym offers two types of memberships: a basic membership for $30 per month and a premium membership for $50 per month. If a customer wants to spend no more than $300 in a year, how many of each type of membership can they purchase?

• Up to 5 basic memberships, or up to 8 premium memberships, or only 2 basic and 4 premium memberships.

• Up to 12 basic memberships, or up to 7 premium memberships, or any combination that exceeds $300.

• Up to 10 basic memberships, or up to 6 premium memberships, or any combination that satisfies 30x + 50y ≤ 300.

• Only 4 basic memberships, or only 3 premium memberships, or a maximum of 2 basic and 5 premium memberships.

A company produces two products, A and B. Each product A requires 3 hours of labor and each product B requires 2 hours of labor. The company has a maximum of 60 hours of labor available. If they want to produce at least 10 products in total, what are the possible combinations of products A and B they can produce?

• (11, -1)

• (5, 6)

• (0, 5)

• Possible combinations: (10, 0), (9, 1), (8, 2), (7, 3), (6, 4), (5, 5), (4, 6), (3, 7), (2, 8), (1, 9), (0, 10)

A restaurant has a seating capacity of 80 people. They serve two types of meals: vegetarian and non-vegetarian. Each vegetarian meal costs $10 and each non-vegetarian meal costs $15. If the restaurant wants to make at least $600 in one night, what combinations of meals can they serve?

• (25, 55)

• Valid combinations of meals include (0, 80), (10, 70), (20, 60), (30, 50), (40, 40), (50, 30), (60, 20), (70, 10), and (80, 0).

• (15, 65)

• (5, 75)

A local theater has 200 seats and sells tickets for $12 for adults and $8 for children. If they want to earn at least $1,500 from ticket sales, what combinations of adult and child tickets can they sell?

• Adult tickets (x) can only be 0 to 25, and child tickets (y) can be calculated as y = 200 + x.

• Adult tickets (x) can range from 50 to 150, and child tickets (y) can be calculated as y = 200 - 2x.

• Adult tickets (x) must be exactly 100, and child tickets (y) can be any number from 0 to 100.

• Adult tickets (x) can range from 25 to 200, and child tickets (y) can be calculated as y = 200 - x.

A charity event is planning to sell two types of tickets: VIP tickets for $100 and regular tickets for $50. If they want to raise at least $2,000 and sell no more than 50 tickets in total, what are the possible combinations of VIP and regular tickets they can sell?

• (25, 0)

• Possible combinations of VIP and regular tickets: (20, 0), (19, 2), (18, 4), (17, 6), (16, 8), (15, 10), (14, 12), (13, 14), (12, 16), (11, 18), (10, 20), (9, 22), (8, 24), (7, 26), (6, 28), (5, 30), (4, 32), (3, 34), (2, 36), (1, 38), (0, 40).

• (30, 10)

• (40, 5)

A factory produces two types of toys: type X and type Y. Each type X toy requires 4 hours of production time and each type Y toy requires 2 hours. If the factory has 80 hours of production time available and wants to produce at least 10 toys, what combinations of type X and type Y toys can they produce?

• Possible combinations are: (0, 10), (1, 8), (2, 6), (3, 4), (4, 2), (5, 0)

• (6, 0)

• (2, 8)

• (0, 5)

A clothing store sells shirts for $25 and pants for $40. If the store wants to make at least $1,000 in sales and has a maximum of 30 items to sell, what combinations of shirts and pants can they offer?

• (0, 30)

• (10, 20)

• (15, 15)

• Combinations include (0, 25), (5, 20), (10, 15), (15, 10), (20, 5), (25, 0) and other variations where x + y <= 30.

A sports club has a budget of $1,200 for new equipment. They can buy basketballs for $30 each and soccer balls for $20 each. If they want to buy at least 20 balls in total, what combinations of basketballs and soccer balls can they purchase?

• (25, 5)

• (15, 10)

• Possible combinations include (0, 60), (10, 30), (20, 0), (1, 58), (2, 56), ..., (40, 0) where x is the number of basketballs and y is the number of soccer balls.

• (5, 45)