Problem 24 - Discrete Mathematics with Applications

Question

Three men and their wives were given \(\$ 5400 .\) The wives together received \(\$ 2400 .\) Sue had \(\$ 200\) more than Jan, and Lynn had \(\$ 200\) more than Sue. Lou got half as much as his wife, Bob the same as his wife, and Matt twice as much as his wife. Who is married to whom? (Mathematics Teacher, 1986)

Step-by-Step Solution

Answer

Lou is married to Sue, Bob is married to Jan, and Matt is married to Lynn.

Step 1: Assign Variables

Let's assign the variables for the wives: let Jan's amount be \(x\), Sue's amount be \(y\), and Lynn's amount be \(z\). Then, Lou's amount will be \(\frac{y}{2}\), Bob's amount will be \(x\), and Matt's amount will be \(2z\).

Step 2: Set up Equations

Now, let's set up the equations based on the information given in the problem: 1. The total amount received by the wives: \(x + y + z = 2400\) 2. Sue had $200 more than Jan: \(y = x + 200\) 3. Lynn had $200 more than Sue: \(z = y + 200\) 4. The total amount received by everyone: \((x+y+z)+\left(\frac{y}{2}+x+2z\right)=5400\)

Step 3: Simplify Equations

Using the information from Equations 2 and 3, we can simplify the system of equations: Equation 1: \(x + y + z = 2400\) Equation 2: \(y = x + 200\) Equation 3: \(z = y + 200\) Equation 4: \((x+y+z)+\left(\frac{y}{2}+x+2z\right)=5400\)

Step 4: Solve the System of Equations

We can now solve the system of equations to find the values of x, y, and z: Substitute Equation 2 into Equation 3: \(z = (x + 200) + 200\) \(z = x + 400\) Now substitute Equations 2 and 3 into Equation 1: \(x + (x + 200) +(x + 400)=2400\) \(3x + 600 = 2400\) Solve for x: \(3x = 1800\) \(x = 600\) Now substitute the value of x back into Equations 2 and 3 to find y and z: \(y = 600 + 200 = 800\) \(z = 800 + 200 = 1000\) Now that we have the amounts each wife received, we can find the amounts for the husbands: Lou: \(\frac{y}{2} = \frac{800}{2} = 400\) Bob: \(x = 600\) Matt: \(2z = 2(1000) = 2000\)

Step 5: Determine the couples

Finally, from the solved amounts, we can determine the couples: - Lou's wife got \(\$800\) , which means Lou is married to Sue. - Bob's wife got \(\$600\), which means Bob is married to Jan. - Matt's wife got \(\$1000\), which means Matt is married to Lynn.

Key Concepts

Systems of Equations

In discrete mathematics, a significant concept is the use of systems of equations to find unknown values. Systems of equations are sets of equations with multiple variables that are solved together. In our exercise, we have three unknowns concerning the amounts of money that Jan, Sue, and Lynn received. The primary goal is to use the information provided to create a system that will allow us to solve for these unknowns.

Here's a brief overview of how such a system can be formed:

• Start by assigning variables for each unknown like amounts received by Jan (\(x\)), Sue (\(y\)), and Lynn (\(z\)).
• Create equations based on relationship statements such as the total amount received by the wives and individual comparisons.
• Combine these equations into a single system to solve all variables simultaneously.

Using systems of equations requires careful setup to ensure all conditions are accurately translated into algebraic form. Once you have the equations, they can reveal much about the relationships between the unknowns, making it easier to deduce additional information like the pairings of husbands and wives.

Algebraic Equations

Algebraic equations form the backbone of solving the problem of distributing amounts and determining relationships between variables. An algebraic equation is a mathematical statement that uses variables alongside constants, linked by operations.

The problem presented can be broken down into basic algebraic steps:

• Initially, translate the verbal statements into mathematical equations, such as expressing Sue's amount as \(y = x + 200\), as she has $200 more than Jan.
• Next, reuse this relationship to express Lynn's amount and ultimately form equations like \(z = y + 200\).
• By combining these equations, we simplify the system, resulting in a solvable set of equations, such as \(3x + 600 = 2400\).

Once you have the equations set, the process of algebra simplifies as you substitute and solve. Always remember each step in the process depends on maintaining balance within the equations to find the correct values for each variable.

Mathematical Reasoning

Mathematical reasoning is about more than just solving equations—it's the cognitive process of understanding how the relationships between the variables interact and lead to a solution. In our example, reasoning is critical to align variables correctly and ensures all equations accurately reflect the problem's conditions.

Here’s how mathematical reasoning comes into play:

• Identify all relationships mentioned in the problem. Recognize coupled pairs by understanding how husband-wife money relationships translate into equations.
• Use logical deduction to connect equations. For instance, from Sue's \(y = x + 200\) and Lynn's subsequent relationship with Sue, establish \(z = y + 200\) effectively.
• Evaluate and validate each step. Finally, test the solutions by ensuring they satisfy all original conditions, proving the relationships fit within the broader problem context.

By employing mathematical reasoning, the problem not only becomes an exercise in computation but also a test of understanding the structure and logic of mathematical problems, which is crucial for grasping more complex discrete math concepts.