Problem 102 - Algebra and Trigonometry

Question

The group should write four different word problems that can be solved using a system of linear equations in two variables. All of the problems should be on different topics. The group should turn in the four problems and their algebraic solutions.

Step-by-Step Solution

Answer

1. Problem: The admission fee at an amusement park is $15 for children and $20 for adults. On a certain day, 220 people enter the park, and $4,300 is collected. How many children and how many adults attended? Solution: Children=70, Adults=150.\n\n2. Problem: A school is selling tickets for a grand concert. The first day of ticket sales, they sold 3 senior tickets and 13 child tickets for a total of $230. The second day, they sold 9 senior tickets and 2 child tickets for a total of $340. How much does each senior and child ticket cost? Solution: Senior=$20, Child=$10.\n\n3. Problem: A shop is selling two kinds of chocolates. A customer buys 3 chocolates of the first kind and 5 chocolates of the second kind for $120. Another customer buys 5 chocolates of the first kind and 2 chocolates of the second kind for $130. What is the cost of each kind of chocolate? Solution: First Kind=$20, Second Kind=$10.\n\n4. Problem: Jack and Jill are saving money. Jack saves $10 a week, and Jill saves $15 a week. After x weeks, Jack has saved a total of $100, and Jill has saved a total of $200. What is the value of x? Solution: x=10 weeks.

Step 1: Formulate the Problems

Think about different real-life scenarios that fit the problem requirements. The scenarios should involve two interrelated variables that could be represented and solved through a system of linear equations. Here are some examples:\n\n1. The admission fee at an amusement park is $15 for children and $20 for adults. On a certain day, 220 people enter the park, and $4,300 is collected. How many children and how many adults attended?\n\n2. A school is selling tickets for a grand concert. The first day of ticket sales, they sold 3 senior tickets and 13 child tickets for a total of $230. The second day, they sold 9 senior tickets and 2 child tickets for a total of $340. How much does each senior and child ticket cost?\n\n3. A shop is selling two kinds of chocolates. A customer buys 3 chocolates of the first kind and 5 chocolates of the second kind for $120. Another customer buys 5 chocolates of the first kind and 2 chocolates of the second kind for $130. What is the cost of each kind of chocolate?\n\n4. Jack and Jill are saving money. Jack saves $10 a week, and Jill saves $15 a week. After x weeks, Jack has saved a total of $100, and Jill has saved a total of $200. What is the value of x?

Step 2: Formulate Systems of Equations

For each of the problems you've come up with, formulate a system of linear equations representing the scenario. Here are the systems for the scenarios proposed:\n\n1. \(c + a = 220, 15c + 20a = 4,300\) where c is the number of children, and a is the number of adults.\n\n2. \(3s + 13c = 230, 9s + 2c = 340\) where s is the price of a senior ticket, and c is the price of a child ticket.\n\n3. \(3c1 + 5c2 = 120, 5c1 + 2c2 = 130\) where c1 is the cost of the first kind of chocolate, and c2 is the cost of the second kind of chocolate.\n\n4. \(10x = 100, 15x = 200\) where x is the number of weeks.

Step 3: Solve The Systems of Equations

Using substitution or elimination, solve each system of equations. Add these solutions to the final problems. Here are the solutions for the systems proposed:\n\n1. Solving the system gives \(c = 70, a = 150\).\n\n2. Solving the system gives \(s = 20, c = 10\).\n\n3. Solving the system gives \(c1 = 20, c2 = 10\).\n\n4. Solving the system gives \(x = 10\).