Understanding the Age Word Problem

This problem involves finding the current age of Ram based on two pieces of information relating his age and his father's age at two different points in time: the present and eight years ago. Age problems are often solved by setting up equations based on the given relationships and then solving those equations.

Setting up the Equations for Age Calculation

Let's use variables to represent the current ages:

• Let Ram's current age be $R$ years.
• Let Ram's father's current age be $F$ years.

Now, we translate the given information into mathematical equations:

1. "Ram’s father is twice as old as Ram is."

   This gives us the first equation:

   $$F = 2R \quad (Equation \; 1)$$

2. "Eight years ago, the age of Ram’s father was 2.5 times his age."

   First, let's determine their ages eight years ago:

   • Ram's age 8 years ago was $R - 8$ years.
   • Ram's father's age 8 years ago was $F - 8$ years.

   Now, set up the relationship given for their ages eight years ago:

   $$F - 8 = 2.5 \times (R - 8) \quad (Equation \; 2)$$

Solving for Ram's Current Age

We now have a system of two linear equations with two variables ($R$ and $F$). We can use substitution to solve for $R$.

Substitute the expression for $F$ from Equation 1 ($F = 2R$) into Equation 2:

$$ (2R) - 8 = 2.5 \times (R - 8) $$

Now, let's solve this equation for $R$ step-by-step:

Expand the right side of the equation:

$$ 2R - 8 = 2.5R - 2.5 \times 8 $$

Calculate $2.5 \times 8$:

$$ 2.5 \times 8 = \left(\frac{5}{2}\right) \times 8 = 5 \times 4 = 20 $$

Substitute this value back into the equation:

$$ 2R - 8 = 2.5R - 20 $$

Gather the terms with $R$ on one side and the constant terms on the other side. Subtract $2R$ from both sides:

$$ -8 = 2.5R - 2R - 20 $$

$$ -8 = 0.5R - 20 $$

Add 20 to both sides:

$$ -8 + 20 = 0.5R $$

$$ 12 = 0.5R $$

To find $R$, divide both sides by 0.5 (which is the same as multiplying by 2):

$$ R = \frac{12}{0.5} $$

$$ R = 12 \times 2 $$

$$ R = 24 $$

So, Ram's current age is 24 years.

Verification of the Age Solution

Let's check if this age satisfies the original conditions:

• Ram's current age $R = 24$ years.
• Father's current age $F = 2R = 2 \times 24 = 48$ years.

Now check the condition for eight years ago:

• Ram's age 8 years ago = $24 - 8 = 16$ years.
• Father's age 8 years ago = $48 - 8 = 40$ years.

Was the father's age 2.5 times Ram's age 8 years ago?

Calculate $2.5 \times (\text{Ram's age 8 years ago})$:

$$ 2.5 \times 16 = \left(\frac{5}{2}\right) \times 16 = 5 \times 8 = 40 $$

The father's age 8 years ago was indeed 40 years, which matches $2.5$ times Ram's age (16 years) at that time. The solution is verified.

The calculated current age of Ram is 24 years.

Revision Table: Key Concepts in Age Problems

Additional Information on Solving Algebraic Word Problems

Age problems are a common type of word problem that can be solved using algebra. Here are some general tips for approaching such problems:

• Read Carefully: Understand the relationships and the different points in time mentioned in the problem.
• Define Variables: Assign variables to the unknown quantities, usually the current ages.
• Formulate Equations: Translate each statement in the problem into a mathematical equation using the defined variables. Pay attention to phrases like "is", "was", "will be", "times", "older than", "younger than".
• Solve the System: Use algebraic methods (like substitution or elimination) to solve the system of equations you created.
• Answer the Question: Make sure your final answer directly addresses what the question is asking for.
• Verify Your Answer: Plug your solution back into the original word problem (not just the equations) to ensure it makes sense and satisfies all conditions.

Practicing different types of word problems, including age problems, helps in developing skills to translate verbal information into mathematical models.