How to Deciphering the Puzzle of Time: A Step-by-Step Guide to Solving Age Problems in Mathematics

Step-by-step Guide to Deciphering Age Problems in Mathematics

Step 1: Understand the Problem

• Read Carefully: Carefully read the problem to understand what you are asked to find.
• Identify Key Information: Look for information about the current ages of individuals, their ages at different times in the past or future, and the relationships between these ages.

Step 2: Define Variables

• Choose Variables: Typically, let \(x\) represent the current age of a person of interest. Ensure that you clearly define what \(x\) represents.
• Represent Other Ages: If the problem involves the person’s age in the past or future, express these ages in terms of \(x\). For example, a person’s age \(a\) years ago is \(x−a\), and their age \(a\) years in the future is \(x+a\).

Step 3: Set Up Equations

• Use the Given Information: Based on the problem’s information, create one or more equations. These equations might relate the ages of different people or describe how a person’s age changes over time.
• Be Consistent: Ensure that the ages are consistent with the timeline given in the problem.

Step 4: Solve the Equations

• Manipulate the Equation: Use algebraic methods to solve for \(x\). This might involve simplifying expressions, combining like terms, or using methods like substitution or elimination in the case of multiple equations.
• Find the Age(s): Solve for \(x\) and any other unknowns in your equations.

Step 5: Verify Your Solution

• Check for Accuracy: Substitute your solution back into the original equations to ensure they hold true.
• Ensure Reasonability: Make sure the solution makes sense in the context of the problem. For instance, ages should be realistic and non-negative.

By following these steps methodically, you can solve a variety of age problems, understanding the relationships between ages at different times.

Examples:

Example 1:

John is currently twice as old as his brother Peter. Five years ago, John was three times as old as Peter. How old are John and Peter now?

Solution:

• Let Peter’s current age be \(x\) years.
• John’s current age is \(2x\) years.
• Five years ago, Peter was \(x−5\) and John was \(2x−5\).
• The equation from the problem:\(2x−5=3(x−5)\).
• Solve: \(2x−5=3x−15\) leads to \(x=10\).
• John is \(2×10=20\) years old.

Peter is \(10\) years old, and John is \(20\) years old.

Example 2:

A mother is four times as old as her daughter. In \(20\) years, she will be twice as old as her daughter. How old are they now?

Solution:

• Let the daughter’s age be \(x\).
• The mother’s age is \(4x\).
• In \(20\) years, the daughter will be \(x+20\) and the mother \(4x+20\).
• The future age equation: \(4x+20=2(x+20)\).
• Solve: \(4x+20=2x+40\) leads to \(x=10\).
• The mother is \(4×10=40\) years old.

The daughter is \(10\) years old, and the mother is \(40\) years old.