Problem 30 - Prealgebra

Question

Diane is 23 years older than her daughter Amy. In 5 years, the sum of their ages will be 91. How old are they now? $$\begin{array}{|l|l|} \hline \underline{\phantom{xxx}} & \hline \text { Now } & \text { In 5 Years } \\ \hline \text { Amy } & x & \\ \hline \text { Diane} & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Amy is 29 years old and Diane is 52 years old.

Step 1: Define the Variables

Let Amy's current age be represented by \( x \). Since Diane is 23 years older than Amy, Diane's current age can be represented as \( x + 23 \).

Step 2: Express Future Ages

In 5 years, Amy's age will be \( x + 5 \) and Diane's age will be \( (x + 23) + 5 = x + 28 \).

Step 3: Set Up the Equation

According to the problem, the sum of Amy's and Diane's ages in 5 years will be 91. Thus, the equation will be: \( (x + 5) + (x + 28) = 91 \).

Step 4: Simplify the Equation

Combine like terms in the equation: \( x + 5 + x + 28 = 91 \) becomes \( 2x + 33 = 91 \).

Step 5: Solve for \( x \)

Subtract 33 from both sides to get \( 2x = 58 \). Then divide both sides by 2 to find \( x \): \( x = 29 \).

Step 6: Calculate Ages

Amy's age is \( x = 29 \) and Diane's age is \( x + 23 = 29 + 23 = 52 \).

Key Concepts

Defining Variables

Defining variables is the fundamental step in solving any word problem, especially age-related ones. By assigning letters or symbols to represent unknown quantities, we create a clear and structured way to work through the problem. In this exercise, we begin by defining Amy's current age as \( x \).

• This simplifies the problem by giving the unknown a name.
• It also allows us to easily express related quantities, such as Diane's age, using that variable.

Given that Diane is 23 years older than Amy, Diane's age can be expressed as \( x + 23 \). This method effectively breaks down the relationship between their ages into manageable parts that can be analyzed mathematically. The key to defining variables is consistency and clarity; it ensures every part of the problem is understood and correctly represented.

Linear Equations

In age problems, linear equations play a crucial role in establishing the relationship between the defined variables and the given conditions. Once you've defined your variables, the next step is to set up a linear equation using these variables.Here, we looked five years into the future. For Amy, her future age is expressed as \( x + 5 \), and Diane's future age becomes \( x + 28 \). By stating that the sum of these ages is 91, we form the linear equation: \[ (x + 5) + (x + 28) = 91 \] Linear equations help in modeling the problem mathematically. By transforming worded conditions into equations, we provide a systematic way to solve for unknowns. The beauty of linear equations in such problems is their simplicity; they can be solved using straightforward algebraic manipulations.

Solving Equations

Solving equations is the step where our defined variables and established equations come to fruition. Here, it's about applying mathematical operations to find the value of \( x \).We start by simplifying the equation: \[ x + 5 + x + 28 = 91 \] Combining like terms gives us the simplified equation:\[ 2x + 33 = 91 \] To isolate \( x \), subtract 33 from both sides:\[ 2x = 58 \] Finally, divide by 2 to solve for \( x \):\[ x = 29 \] Finding \( x \) means we've determined Amy's current age. Using Amy's age, we then find Diane's age by calculating \( x + 23 = 52 \). This method of solving involves understanding each step logically—simplification, isolation, and verification, which together give you clear answers from complex scenarios.