Problem 31 - Prealgebra

Question

Dale is 4 years older than Sue. Five years ago the sum of their ages was 64. How old are they now?

Step-by-Step Solution

Answer

Sue is 35 years old, and Dale is 39 years old.

Step 1: Define Variables

Let Dale's current age be \( d \) and Sue's current age be \( s \). We know that Dale is 4 years older than Sue, so we can express this relationship as \( d = s + 4 \).

Step 2: Set Up Equation

According to the problem, five years ago, the sum of their ages was 64. Thus, we can write the equation: \((d - 5) + (s - 5) = 64\).

Step 3: Simplify the Equation

Simplify the equation from Step 2: \((d - 5) + (s - 5) = 64\) simplifies to \(d + s - 10 = 64\). Further simplification gives \(d + s = 74\).

Step 4: Substitute and Solve

Substitute \(d = s + 4\) from Step 1 into \(d + s = 74\). This results in: \((s + 4) + s = 74\), which simplifies to \(2s + 4 = 74\).

Step 5: Solve for Sue's Age

Solve the equation \(2s + 4 = 74\). Subtract 4 from both sides: \(2s = 70\), then divide by 2: \(s = 35\). So, Sue is 35 years old.

Step 6: Solve for Dale's Age

Substitute \(s = 35\) into \(d = s + 4\) to find Dale's age. Thus, \(d = 35 + 4 = 39\). Therefore, Dale is 39 years old.

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that provides a way to represent and solve problems using symbols and letters. In many word problems, including age-related ones, algebra allows us to create equations that model real-world situations. It provides a structured approach to finding unknown values. When we encounter a problem like the one given with Dale and Sue, we use algebraic techniques to express relationships. The key to success with algebra is understanding how to translate from words into mathematical expressions. This involves recognizing patterns and relationships described in the problem, translating those into algebraic terms, and then manipulating them according to mathematical rules.
Algebra helps us see these relationships in a structured, logical manner. Symbols like variables stand in for numbers, enabling us to set up equations that capture the essential elements of the problem. Once set up, these equations can be solved systematically, providing precise answers.

Linear Equations

Linear equations are a type of equation that involves variables raised only to the first power, resulting in solutions that form a straight line when graphed. In age problems like our example with Dale and Sue, we often deal with linear equations because we are simply adding, subtracting, or equating expressions involving the variables.
Linear equations might look like this: \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. In our problem, we formed a linear equation based on the relationship between Dale and Sue's ages: \( d + s = 74 \). This is derived by combining given information and simplifications.
This simple form makes solving for unknowns straightforward. We can isolate variables on one side of the equation and perform arithmetic operations to find their values. Understanding linear equations is crucial because it allows us to break down complex real-world problems into simpler, manageable components.

Variable Substitution

Variable substitution is a technique used in algebra to solve equations by replacing one variable with an expression involving another variable. This method is particularly useful when dealing with multiple equations, as seen in our example.
When we defined Dale's age with the variable \( d \) as \( d = s + 4 \), we created a substitution that simplifies the problem significantly. Using this substitution, we replaced \( d \) in the equation \( d + s = 74 \) with \( s + 4 \). This gave us a single equation in terms of one variable: \( 2s + 4 = 74 \).
This process of substitution reduces the number of variables we deal with and converts complex multiple-variable problems into simpler single-variable ones. By isolating and solving for one variable first, we can then easily find the other variable's value by substitution back into one of the equations. Mastering variable substitution is essential for tackling many algebraic problems, especially when equations inter-relate, as is common in word problems.