Problem 32 - Prealgebra

Question

Pat is 2 years younger than his wife, Wynn. Ten years ago the sum of their ages was 48 . How old are they now?

Step-by-Step Solution

Answer

Wynn is 35 years old and Pat is 33 years old.

Step 1: Define the Variables

Let Wynn's current age be \( x \). Since Pat is 2 years younger than Wynn, Pat's current age is \( x - 2 \).

Step 2: Establish the Equation for 10 Years Ago

Ten years ago, Wynn's age was \( x - 10 \) and Pat's age was \( x - 2 - 10 = x - 12 \). We're given that the sum of their ages ten years ago was 48, so we can write the equation: \((x - 10) + (x - 12) = 48\).

Step 3: Simplify the Equation

Combine like terms in the equation: \((x - 10) + (x - 12) = 48\) simplifies to \(2x - 22 = 48\).

Step 4: Solve for Wynn's Current Age

Add 22 to both sides of the equation: \(2x - 22 + 22 = 48 + 22\), resulting in \(2x = 70\). Divide both sides by 2 to solve for \(x\): \(x = 35\). So, Wynn is currently 35 years old.

Step 5: Calculate Pat's Current Age

Since Pat is 2 years younger than Wynn, Pat's age is \(35 - 2 = 33\). Thus, Pat is currently 33 years old.

Key Concepts

Equations

An equation is like a sentence in math, with an equal sign as the verb. Equations show how two expressions are equivalent, they balance both sides. For example, we used

• "\((x - 10) + (x - 12) = 48\)"

for our problem. Here, the left side represents Wynn and Pat's ages ten years ago. The right side is their combined age at that time.
Creating equations helps translate word problems into math, simplifying them to find solutions. By understanding and setting up equations effectively, we solve problems and find unknown values.

Age Difference

Age difference is simply the gap in years between two people's ages. It helps understand relationships in age word problems.
In our exercise, Pat is 2 years younger than Wynn. The age difference remains constant over time. By using this constant difference, problems become simpler and structured. age difference helps:

• Define relationships
• Establish equations
• Predict future ages or check past ages

By knowing the age difference and one person's age, you can easily find the other's. It's crucial to consider it to avoid confusion while solving these problems.

Solving Equations

Solving equations involves finding the unknown variable's value that makes the equation true. Follow these basic steps to solve an equation:

• Combine like terms: Make the equation simpler by adding or subtracting similar terms on the same side.
• Isolate the variable: Use basic operations like addition, subtraction, multiplication, or division to move other terms away from the variable until it stands alone.

In our example, after simplifying "\(2x - 22 = 48\)", we add 22 to both sides. It gives "\(2x = 70\)", then divide by 2 to isolate \(x\) (Wynn's age): "\(x = 35\)". Breaking down each step and staying organized makes equations easier to tackle, leading to correct solutions.