Understanding the Ram's Age Problem

This question asks us to find Ram's current age based on given relationships between his age and his father's age at two different points in time: currently and 4 years ago. These types of problems are classic age word problems that can be solved using algebraic equations.

Setting Up the Age Problem

To solve this age problem, we can use variables to represent the unknown ages. Let's define:

• Let Ram's current age be \(R\) years.
• Let Ram's father's current age be \(F\) years.

Formulating Equations from the Problem Statement

The problem gives us two key pieces of information, which we can translate into two equations:

1. "Ram’s father is thrice as old as Ram is." This describes their current ages.
   In terms of our variables, this means:
   \(F = 3R\)
2. "4 years ago, the age of Ram’s father was 4 times his age." This describes their ages in the past.
   4 years ago, Ram's age was \(R - 4\) years.
   4 years ago, Ram's father's age was \(F - 4\) years.
   The relationship given is that the father's age 4 years ago was 4 times Ram's age 4 years ago:
   \(F - 4 = 4 \times (R - 4)\)

Now we have a system of two linear equations with two variables:

• Equation 1: \(F = 3R\)
• Equation 2: \(F - 4 = 4(R - 4)\)

Solving the System of Equations to Find Ram's Current Age

We can use substitution to solve this system. We already have an expression for \(F\) in terms of \(R\) from Equation 1. We can substitute \(3R\) for \(F\) into Equation 2.

1. Substitute \(F = 3R\) into Equation 2:
   \( (3R) - 4 = 4(R - 4) \)
2. Now, simplify and solve for \(R\):
   Distribute the 4 on the right side of the equation:
   \( 3R - 4 = 4R - 16 \)
   Move the terms involving \(R\) to one side and the constant terms to the other side. Subtract \(3R\) from both sides:
   \( -4 = 4R - 3R - 16 \)
   \( -4 = R - 16 \)
   Add 16 to both sides to isolate \(R\):
   \( -4 + 16 = R \)
   \( 12 = R \)

So, Ram's current age is 12 years.

We can check this answer using the original equations:

• If Ram's current age (\(R\)) is 12, his father's current age (\(F\)) should be \(3 \times 12 = 36\) years. (Checks Equation 1: \(36 = 3 \times 12\))
• 4 years ago, Ram's age was \(12 - 4 = 8\) years.
• 4 years ago, his father's age was \(36 - 4 = 32\) years.
• Is the father's age 4 years ago (32) 4 times Ram's age 4 years ago (8)?
  \(32 = 4 \times 8\)
  \(32 = 32\) (Checks Equation 2)

Both conditions are satisfied, confirming that Ram's current age is 12 years.

Conclusion: Ram's Current Age

Based on our calculations from the given information, Ram's current age is 12 years.

Revision Table: Key Concepts for Age Problems

Additional Information: Solving Age Word Problems

Age word problems are a common application of linear equations. They often involve relationships between the ages of two or more people at different points in time. The key to solving them is careful translation of the word phrases into mathematical expressions and equations.

• Phrases like "twice as old as" or "thrice as old as" indicate multiplication.
• Phrases like "X years ago" require subtracting X from the current age.
• Phrases like "in X years" or "after X years" require adding X to the current age.
• The relationships given in the problem usually provide the equations needed to solve for the variables.

Always define your variables clearly and state what they represent (usually current ages). This helps avoid confusion when setting up the equations and interpreting the final answer.