Solving Two-Variable Age Problems: A Step-by-Step Guide

By: Justin 3/9/2025

Understanding Two-Variable Age Problems

Two-variable age problems typically involve two people (e.g., a parent and a child, two siblings, or two friends) and describe how their ages relate to each other at different points in time. Your goal is to set up two equations based on the given conditions and solve for their ages.

Common Clues in Age Problems

When solving age problems, pay attention to the following common phrases:

  • “Twice as old” or “Three times as old” → This suggests a multiplication relationship.
  • “Three years older” or “Ten years older” → This suggests a sum relationship.
  • “Three years younger” or “Ten years younger”→ This suggests a difference relationship.
  • “X years ago” or “X years from now” → This indicates a time shift, meaning you must adjust both ages accordingly.
  • “Sum of their ages” → This provides a direct equation where adding both ages results in a given total.
  • “Difference of their ages” → This suggests a subtraction equation.

Step-by-Step Approach to Solving Two-Variable Age Problems

Step 1: Define Your Variables

Let:

  • \( x \) = the age of one person
  • \(y \) = the age of the second person

Step 2: Create Two Equations

Use the given relationships in the problem to form two equations.

Step 3: Solve the Equations

Use either substitution or elimination to solve for \(x\) and \( y\).

Step 4: Verify Your Answer

Plug your values back into the original conditions to ensure correctness.

Example 1: Present Age Relationship

Problem:
Mark is twice as old as John. The sum of their ages is 36 years. How old are Mark and John?

Step 1: Define Variables

Let:

  • \( x \) = Mark’s age
  • \(y \) = John’s age

Step 2: Set Up Equations

From the problem statement:

  1. Mark is twice John’s age:
    \( x = 2y \)
  2. The sum of their ages is 36:
    \( x + y = 36 \)

Step 3: Solve the Equations

Substituting \( x = 2y \) into the second equation:
\( 2y + y = 36 \)

\( 3y = 36 \)

\( y = 12 \)

Now, solving for \( x \):
\(x = 2(12) = 24\)

Final Answer:

Mark is 24 years old, and John is 12 years old.

Example 2: Age Relationship in the Past

Problem:
A father is two times as old as his son. Ten years ago, he was three times as old. How old are they now?

Step 1: Define Variables

Let:

  • \( x \) = Father’s current age
  • \( y \) = Son’s current age

Step 2: Set Up Equations

From the given information:

  1. The father is two times as old as the son:
    \(x = 2y\)

  2. Ten years ago, the father was three times as old as the son:
    \((x - 10) = 3(y - 10)\)

Step 3: Solve the Equations

Substituting \( x = 2y \) into the second equation:
\( (2y - 10) = 3(y - 10) \)

Expanding:
\( 2y - 10 = 3y - 30 \)

Rearrange the equation:
\(2y - 3y = -30 + 10\)

\(-y = -20\)

\(y = 20\)

Now, solving for \( x \):

\(x = 2(20) = 40\)

Final Answer:

The father is 40 years old, and the son is 20 years old.

Example 3: Age Relationship in the Future

Problem:
Anna is 5 years older than Ben. In 3 years, the sum of their ages will be 41. How old are they now?

Step 1: Define Variables

Let:

  • \( x \) = Anna’s current age
  • \(y \) = Ben’s current age

Step 2: Set Up Equations

From the given information:

  1. Anna is 5 years older than Ben:
    \(x = y + 5\)

  2. In 3 years, the sum of their ages will be 41:
    \( (x + 3) + (y + 3) = 41 \)

Step 3: Solve the Equations

Expanding the second equation:
\( x + 3 + y + 3 = 41 \)

\( x + y + 6 = 41 \)

\( x + y = 35 \)

Substituting \( x = y + 5\) into \( x + y = 35 \):

\( (y + 5) + y = 35 \)

\( 2y + 5 = 35 \)

\( 2y = 30 \)

\( y = 15 \)

Now, solving for \(x\):
\( x = 15 + 5 = 20 \)

Final Answer:

Anna is 20 years old, and Ben is 15 years old.

Key Takeaways for Solving Two-Variable Age Problems

  • Define your variables clearly based on the people involved.
  • Identify key relationships (e.g., "twice as old," "older than," "sum of their ages").
  • Set up two equations based on the given conditions.
  • Use substitution or elimination to solve for unknowns.
  • Check your final answers by plugging them back into the original conditions.

By mastering these steps, you’ll be able to solve two-variable age problems quickly and efficiently, especially in exams like the Civil Service Exam.

For more mock exams and practice problems, visit brevph and sharpen your problem-solving skills today! 🚀

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