Problem 29 - Prealgebra

Question

Pat is 20 years older than his son Patrick. In 2 years, the sum of their ages will be \(90 .\) How old are they now? $$\begin{array}{|l|l|} \hline \underline{\phantom{xxx}} & \hline \text { Now } & \text { In 2 Years } \\ \hline \text { Patrick } & x & \\ \hline \text { Pat } & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Patrick is 33 years old, and Pat is 53 years old.

Step 1: Define the Variables

Let's denote Patrick's current age as \( x \). Since Pat is 20 years older than Patrick, Pat's current age will be \( x + 20 \).

Step 2: Express Future Ages

In 2 years, Patrick's age will be \( x + 2 \), and Pat's age will be \( x + 20 + 2 \).

Step 3: Set Up the Equation

According to the problem, in 2 years, the sum of their ages will be 90. Thus, we can write the equation as: \[ (x + 2) + (x + 20 + 2) = 90 \]

Step 4: Simplify the Equation

Simplify the left side of the equation: \[ x + 2 + x + 20 + 2 = 90 \] Combine like terms: \[ 2x + 24 = 90 \]

Step 5: Solve the Equation

Subtract 24 from both sides of the equation to isolate terms with \( x \): \[ 2x = 66 \] Divide both sides by 2 to solve for \( x \): \[ x = 33 \]

Step 6: Find Pat's Age

Patrick's current age is \( x = 33 \). Therefore, Pat’s age is \( x + 20 = 33 + 20 = 53 \).

Step 7: Verify the Solution

In 2 years, Patrick will be \( 35 \) and Pat will be \( 55 \). The sum of their ages will indeed be 90, verifying the solution as correct.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two expressions. They typically contain variables, numbers, and arithmetic operations. In the current exercise, we set up an algebraic equation to model the relationship between the ages of Pat and Patrick. The sum of their ages in two years is given as 90, which leads us to construct the equation: \[(x + 2) + (x + 20 + 2) = 90\]This equation represents a real-world scenario using algebra. It captures how changes in time affect the ages of Pat and Patrick. By expressing these relationships algebraically, we can systematically solve for unknown variables, offering a clear pathway to finding their exact ages.

Defining Variables

Defining variables is a foundational step in solving word problems like this one. A variable stands in for an unknown value that we are attempting to determine. In our exercise, the variable \(x\) denotes the current age of Patrick. This allows us to also express Pat's age as \(x + 20\), because Pat is stated to be 20 years older. By defining these variables, we convert a verbal description into an algebraic problem, which is easier to manipulate. It's important to choose variables wisely to simplify and solve the problem efficiently. Defining variables helps us break down complex situations into manageable parts.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. After defining our variables and setting up our equation as \[2x + 24 = 90\]we need to simplify and solve it. First, we isolate terms involving \(x\) by subtracting 24 from both sides, yielding: \[2x = 66\]Then, we divide both sides by 2 to solve for \(x\), giving us \(x = 33\). Once we have Patrick's age, we can derive Pat's age using the relationship \(x + 20 = 53\). Solving equations like these requires performing inverse operations to systematically isolate the variable. Each step brings us closer to the final answer, allowing us to verify our solution by checking the initial conditions were met.