Question

Meg is twice as old as Rolf, but three years ago, she was two years older than Rolf is now. How old is Rolf now?

Short answer

Rolf is 5 years old.

Step-by-step solution

Define the Variables

Let Meg's current age be represented by M and Rolf's current age by R.

Set Up the First Equation

According to the problem, Meg is twice as old as Rolf. Therefore, we can express this relationship as: M = 2R

Set Up the Second Equation

Three years ago, Meg's age can be represented as M - 3 and Rolf's age as R - 3. The problem states that three years ago, Meg was two years older than Rolf is now. Thus, we can set up the following equation: M - 3 = R + 2

Solve the System of Equations

Now, we have a system of equations:1) M = 2R2) M - 3 = R + 2Substitute the value of M from the first equation into the second equation: 2R - 3 = R + 2

Simplify the Equation

Solve for R by simplifying the equation: 2R - 3 = R + 2 Subtract R from both sides: R - 3 = 2 Add 3 to both sides: R = 5

Verify the Solution

Verify the solution by checking the conditions given in the problem: If Rolf is 5 years old, then Meg is twice his age, which is 10 years old. Three years ago, Meg was 10 - 3 = 7 years old. At the same time, Rolf's current age (5) plus 2 is also 7. This confirms our solution is correct.

Key concepts

Defining Variables in Algebra

When solving age-related algebra problems, defining variables is the first step. A variable is a symbol that represents an unknown value. It's often helpful to label the variables based on the information given in the problem.
For example, let Meg's current age be represented by M and Rolf's current age by R. By assigning variables like this, you create a clear roadmap for setting up and solving equations later on.
Remember, the variable names should be intuitive and meaningful to help you understand their role in the problem.

Setting Up Equations

After defining the variables, the next step is to set up equations based on the relationships described in the problem. Equations are mathematical statements that show the equality between two expressions.
In our example, we know that Meg is twice as old as Rolf. Therefore, we can represent this relationship with the equation: \( M = 2R \).
The problem also provides another piece of information: three years ago, Meg was two years older than Rolf is now. We can translate this into another equation: \( M - 3 = R + 2 \).
By translating the word problem into mathematical equations, we create a system of equations that can be solved.

Solving Systems of Equations

With the equations set up, the next step is to solve the system of equations. This involves finding the values of the variables that satisfy both equations simultaneously.
Our system of equations is: \( M = 2R \) and \( M - 3 = R + 2 \).
To solve this system, we can use substitution or elimination.
For this example, we'll use substitution. We know that \( M = 2R \) from the first equation. We can substitute \( 2R \) in place of \( M \) in the second equation: \( 2R - 3 = R + 2 \).
By simplifying this equation, we get: \( 2R - R - 3 = 2 \), which simplifies to: \( R - 3 = 2 \).
Finally, add 3 to both sides to find \( R \): \( R = 5 \).

Verifying Algebraic Solutions

Once we have a solution, it's important to verify it to ensure it's correct. Verification involves plugging the solution back into the original problem to check if it satisfies all given conditions.
In this example, we found that Rolf is 5 years old. According to our first equation, Meg would be twice Rolf's age: \( M = 2 * 5 = 10 \) years old.
Three years ago, Meg's age was: \( 10 - 3 = 7 \) years. Now, check if this satisfies the second condition: Rolf's current age plus 2 should be: \( 5 + 2 = 7 \) years.
Since both conditions are met, our solution is verified. Always take this final step to ensure your solution is accurate and satisfies the original problem's requirements.