Solving Age Word Problems: Finding Current Ages

This problem involves finding the current ages of three individuals, A, B, and C, based on relationships between their ages at different points in time. We are given three conditions that relate their ages.

Defining Variables for Ages

Let's represent the current ages of A, B, and C with variables:

• Let the current age of A be \(A\) years.
• Let the current age of B be \(B\) years.
• Let the current age of C be \(C\) years.

Setting Up Equations from Given Conditions

We can translate the given information into mathematical equations:

1. "Before 4 years, the age of A was twice the age of B."

   4 years ago, A's age was \(A - 4\). 4 years ago, B's age was \(B - 4\). The condition states \( (A - 4) = 2 \times (B - 4) \). This simplifies to: \(A - 4 = 2B - 8\) \(A = 2B - 8 + 4\) \(A = 2B - 4\) (Equation 1)

2. "The current age of A is 10 years more than that of B."

   This directly translates to: \(A = B + 10\) (Equation 2)

3. "After 6 years, the age of C will be twice the age of A."

   After 6 years, C's age will be \(C + 6\). After 6 years, A's age will be \(A + 6\). The condition states \( (C + 6) = 2 \times (A + 6) \). This simplifies to: \(C + 6 = 2A + 12\) \(C = 2A + 12 - 6\) \(C = 2A + 6\) (Equation 3)

Solving for Current Ages Step-by-Step

We now have a system of three equations with three variables. We can solve this system to find the current ages.

Step 1: Find the current ages of A and B

We have two equations relating A and B:

• Equation 1: \(A = 2B - 4\)
• Equation 2: \(A = B + 10\)

Since both equations are equal to A, we can set them equal to each other:

\(2B - 4 = B + 10\)

Now, solve for B:

\(2B - B = 10 + 4\)

\(B = 14\)

So, the current age of B is 14 years.

Now substitute the value of B into either Equation 1 or Equation 2 to find A. Using Equation 2 is simpler:

\(A = B + 10\)

\(A = 14 + 10\)

\(A = 24\)

So, the current age of A is 24 years.

Step 2: Find the current age of C

Now that we know the current age of A (which is 24 years), we can use Equation 3 to find the current age of C:

• Equation 3: \(C = 2A + 6\)

Substitute the value of A into Equation 3:

\(C = 2 \times 24 + 6\)

\(C = 48 + 6\)

\(C = 54\)

So, the current age of C is 54 years.

Summary of Current Ages

Based on our calculations:

• Current age of A: 24 years
• Current age of B: 14 years
• Current age of C: 54 years

Let's quickly check if these ages satisfy the original conditions:

• Before 4 years: A was \(24-4=20\), B was \(14-4=10\). Is 20 twice 10? Yes, \(20 = 2 \times 10\). (Condition 1 satisfied)
• Current ages: A is 24, B is 14. Is 24 10 more than 14? Yes, \(24 = 14 + 10\). (Condition 2 satisfied)
• After 6 years: C will be \(54+6=60\), A will be \(24+6=30\). Will 60 be twice 30? Yes, \(60 = 2 \times 30\). (Condition 3 satisfied)

All conditions are satisfied, confirming our calculated ages are correct.

The current age of C is 54 years.

Revision Table: Age Calculation Summary

Additional Information: Mastering Age Problems

Age problems are common in algebra and quantitative aptitude sections of exams. They typically involve relationships between current ages or ages at different points in time (past or future).

• Key Strategy: Always define variables for the current ages. This is the most straightforward approach.
• Time Adjustments:
  • To represent age in the past, subtract the number of years from the current age.
  • To represent age in the future, add the number of years to the current age.
• Formulating Equations: Carefully read each statement in the problem and translate it into an algebraic equation using the defined variables and time adjustments.
• Solving Equations: Age problems usually result in a system of linear equations. Use methods like substitution or elimination to solve for the unknown variables.
• Verification: After finding the ages, always plug them back into the original conditions stated in the problem to ensure they satisfy all requirements. This helps catch potential errors.
• Common Phrases:
  • "X years ago": Current age - X
  • "After Y years" or "In Y years": Current age + Y
  • "Twice the age": \(2 \times\) age
  • "Half the age": \(\frac{1}{2} \times\) age
  • "Age difference": Usually remains constant over time.

Practicing various types of age problems helps in quickly setting up the correct equations and solving them efficiently.