Introduction to Age Problems

Age problems are common in math and often appear in exams like the civil service exam. These problems involve relationships between ages of people at different points in time and usually require forming equations to find the unknown ages.

Common Clues in Age Problems:

1. Current Age: The present age of a person.
2. Past Age: The age of a person at a certain time in the past.
3. Future Age: The age of a person at a certain time in the future.
4. Relationships: Age differences, sum of ages, or multiples of ages.

Basic Approach to Solving Age Problems:

1. Define Variables: Assign variables to represent the unknown ages. For example, let $x$ be the current age of a person.

2. Set Up Equations: Use the information given to set up an equation or equations. Key phrases often used are:

   • “Years ago” indicates subtraction.
   • “Years from now” indicates addition.
   • “Twice as old,” “half as old,” etc., indicates multiplication or division.

3. Solve the Equations: Combine the equations, if necessary, to find the unknown values.

4. Check the Answer: Substitute your solution back into the problem to ensure it fits all conditions.

Example Problem:

Problem:
Anna is twice as old as her brother Mark. In 5 years, the sum of their ages will be 35. How old are they now?

Solution:

1. Define variables:
   Let $x$ be Mark’s current age.
   Then, Anna’s current age is $2x$.

2. Set up the equation based on the future scenario:
   In 5 years, Mark’s age will be $x+5$ and Anna’s age will be $2x+5$.

3. Use the sum of their future ages:

   $$(x+5)+(2x+5)=35$$

4. Simplify and solve:

   $$3x+10=353x=25x=8.33$$

Since ages must be whole numbers, check if rounding fits the condition or interpret the context.

Key Tips:

• Read carefully to identify the current, past, or future context.
• Express all ages in terms of the same variable.
• Check each solution step to ensure it logically follows from the previous information.