Problem 41 - Elementary and Intermediate Algebra

Question

Sharing Raffle Tickets. Hal gives Tom as many raffle tickets as Tom first had and Gary as many as Gary first had. In like manner, Tom then gives Hal and Gary as many tickets as each then has. Similarly, Gary gives Hal and Tom as many tickets as each then has. If each finally has 40 tickets, with how many tickets does Tom begin?

Step-by-Step Solution

Answer

Tom begins with 20 tickets.

Step 1 - Define Variables

Let the initial number of tickets for Hal be H, for Tom be T, and for Gary be G. Initially, Hal has H tickets, Tom has T tickets, and Gary has G tickets.

Step 2 - Hal Gives Tickets

Hal gives Tom T tickets and Gary G tickets. After this exchange, Hal has (H - T - G) tickets, Tom has (T + T = 2T) tickets, and Gary has (G + G = 2G) tickets.

Step 3 - Tom Gives Tickets

Next, Tom gives Hal (H - T - G) tickets and Gary (2G) tickets. After this exchange, Hal has (H - T - G + H - T - G = 2H - 2T - 2G) tickets, Tom has (2T - (H - T - G) - 2G = 3T - H) tickets, and Gary has (2G + 2G = 4G) tickets.

Step 4 - Gary Gives Tickets

Then, Gary gives Hal (2H - 2T - 2G) tickets and Tom (3T - H) tickets. After this exchange, Hal has (2H - 2T - 2G + 2H - 2T - 2G = 4H - 4T - 4G) tickets, Tom has (3T - H + 3T - H = 6T - 2H) tickets, and Gary has (4G - (2H - 2T - 2G) - (3T - H) = 40).

Step 5 - Set Equations Equal to 40

Finally, each person ends up with 40 tickets, so we set up the following equations: 4H - 4T - 4G = 40, 6T - 2H = 40, 4G = 40.

Step 6 - Solve the System of Equations

Solve the equation 4G = 40 to get G = 10. Next, substitute G = 10 into 4H - 4T - 4G = 40 to get 4H - 4T - 40 = 40, simplify to get 4H - 4T = 80, which simplifies further to H - T = 20. Lastly, solve 6T - 2H = 40 for T: simplify to get 3T - H = 20.

Step 7 - Find Tom's Initial Tickets

Now, we have two simpler equations: H - T = 20 and 3T - H = 20. Solving these simultaneously, add the two equations: \[H - T + 3T - H = 20 + 20\], which simplifies to \[2T = 40\], giving T = 20.

Key Concepts

System of Equations

A system of equations refers to two or more equations that share common variables. Understanding how to solve these systems is very useful in algebra.

Imagine you have multiple conditions to fulfill at once, like in our raffle tickets example. Here, each person's final ticket count depends on the exchanges made by everyone.

To solve these equations, we need to find a set of values that satisfies all the conditions simultaneously. This often involves techniques like substitution or elimination, which we will use later in the process.

Variables in Algebra

Variables are symbols (like H, T, G) used to represent unknown numbers in equations.

When solving an algebraic problem, defining your variables clearly is the first step. In the raffle ticket problem, like in our example, we used:

• H for the number of tickets Hal starts with
• T for the number of tickets Tom starts with
• G for the number of tickets Gary starts with

Getting the variables right helps in setting up the equations correctly. Think of them as placeholders that will eventually be replaced with actual numbers once we solve the system.

Step-by-Step Solution

Breaking down a problem into small, manageable steps is key to finding the right solution. Here's how the steps worked in the raffle tickets problem:

• First, define your variables to represent the unknowns.
• Next, create equations based on the actions described. For example, Hal gives away tickets, Tom gives away tickets, and so on.
• Then, express these interactions as equations with your variables.
• Simplify these equations step-by-step until you have a clearer form.

This structured way of approaching a problem helps you to avoid mistakes and keeps the solution process organized.

Equation Solving

Solving equations means finding the values of variables that make the equation true.

In our exercise, we had to solve:

• 4H - 4T - 4G = 40
• 6T - 2H = 40
• 4G = 40

Start by solving the simplest equation first, which was 4G = 40, giving G = 10. Then, substitute back into the other equations to find the values of H and T. This process can involve combining equations or substituting values back and forth.

Remember, solving equations systematically is crucial. Check your final answers to ensure they satisfy all given conditions.