Problem 200 - Elementary Algebra

Question

The age of Mark's dad is 4 less than twice Marks's age. The sum of their ages is ninety-five. Find their ages.

Step-by-Step Solution

Answer

Mark is 33 years old and his dad is 62 years old.

Step 1: Define Variables

Let Mark's age be denoted by \(x\).

Step 2: Express Dad's Age in Terms of Mark's Age

Since Mark's dad is 4 years less than twice Mark's age, his dad’s age can be written as \(2x - 4\).

Step 3: Set Up the Equation

The sum of their ages is given to be ninety-five: \(x + (2x - 4) = 95\).

Step 4: Simplify the Equation

Combine like terms: \(x + 2x - 4 = 95\) simplifies to \(3x - 4 = 95\).

Step 5: Solve for x

Add 4 to both sides: \(3x = 99\). Then divide by 3 to find \(x = 33\).

Step 6: Find Dad's Age

Substitute \(x\) back into the expression for Dad's age: \(2x - 4 \rightarrow 2(33) - 4 = 66 - 4 = 62\).

Key Concepts

defining variables

When solving age problems, the first and most crucial step is to define the variables. Variables are symbols we use to represent unknown values. In this problem, we need to find the ages of Mark and his dad.
By using a variable, we can represent these unknown values with algebraic expressions.
In this case, we let Mark's age be denoted by the variable \(x\). This allows us to describe other related quantities in terms of \(x\).
For example, Mark's dad's age can be expressed using the variable \(x\), making it easier to set up equations and ultimately solve the problem.

setting up equations

Setting up equations involves translating the words of the problem into mathematical expressions. Once we have defined our variables, we need to use the information given in the problem to create equations.
In this exercise, we know that Mark's dad's age is 4 years less than twice Mark's age. This can be written as \(2x - 4\), where \(x\) is Mark's age.
We are also given that the sum of their ages is ninety-five. This means we can write another equation: \(x + (2x - 4) = 95\).
By setting up these equations, we create a system of equations that will help us find the unknown values.

solving linear equations

Once we have our equation set up, the next step is to solve it. In this case, our equation is \(x + (2x - 4) = 95\).
First, we combine like terms to simplify the equation: \(x + 2x - 4 = 95\) simplifies to \(3x - 4 = 95\).
The next step is to isolate the variable \(x\) on one side of the equation. We add 4 to both sides: \(3x - 4 + 4 = 95 + 4\), which simplifies to \(3x = 99\).
Finally, we divide both sides by 3 to find \(x = 33\), which tells us that Mark is 33 years old.

substitution method

After finding the value of \(x\), we use the substitution method to find the value of related variables. The substitution method involves replacing the variable in one equation with the value we found.
In this problem, we need to find Mark's dad's age. We know from our previous work that Mark's dad's age is represented by \(2x - 4\).
We substitute the value of \(x\) that we found earlier into this expression: \(2(33) - 4\).
This simplifies to \(66 - 4 = 62\), giving us Mark's dad's age.
Using the substitution method ensures that we accurately find the value of related variables, completing the problem.