Understanding the Age Word Problem

This question is an example of an age word problem where we are given relationships between the ages of different people and the total of their ages. We need to find the age of one specific person, Goutam, using the given information.

Let's break down the information given:

• Satheesh is two years older than Goutam.
• Goutam is twice as old as Sai.
• The total age of Satheesh, Goutam, and Sai is 27.

Setting Up Variables for Ages

To solve this age word problem, we can use variables to represent the ages. It's often easiest to assign a variable to the person whose age is the base for other relationships. In this case, Goutam's age is defined based on Sai's age, and Satheesh's age is defined based on Goutam's age. So, let's start with Sai's age.

• Let Sai's age be represented by \( S \).
• Goutam is twice as old as Sai. So, Goutam's age \( G = 2 \times S \).
• Satheesh is two years older than Goutam. So, Satheesh's age \( Sa = G + 2 \). Since \( G = 2S \), Satheesh's age \( Sa = 2S + 2 \).

Formulating the Equation

The total of the ages of Satheesh, Goutam, and Sai is 27. We can write this as an equation using our variables:

\( Sa + G + S = 27 \)

Now, substitute the expressions for \( Sa \) and \( G \) in terms of \( S \) into this equation:

\( (2S + 2) + (2S) + S = 27 \)

Solving for Sai's Age

Now we have a linear equation with one variable, \( S \). Let's solve for \( S \):

\( 2S + 2 + 2S + S = 27 \)

Combine the terms with \( S \):

\( (2S + 2S + S) + 2 = 27 \)

\( 5S + 2 = 27 \)

Subtract 2 from both sides of the equation:

\( 5S = 27 - 2 \)

\( 5S = 25 \)

Divide both sides by 5:

\( S = \frac{25}{5} \)

\( S = 5 \)

So, Sai's age is 5 years.

Calculating Goutam's Age

The question asks for Goutam's age. We know that Goutam's age \( G = 2 \times S \). Now that we know \( S = 5 \), we can find Goutam's age:

\( G = 2 \times 5 \)

\( G = 10 \)

Therefore, Goutam is 10 years old.

Verification of Ages

Let's check if these ages satisfy all the conditions given in the problem:

• Sai's age \( S = 5 \)
• Goutam's age \( G = 10 \) (\( 2 \times 5 = 10 \), which is twice Sai's age)
• Satheesh's age \( Sa = G + 2 = 10 + 2 = 12 \) (which is two years older than Goutam)

The total age is \( Sa + G + S = 12 + 10 + 5 = 27 \). This matches the total age given in the question.

All conditions are satisfied, so our calculated ages are correct. Goutam's age is 10.

The final answer is 10.

Revision Table: Age Word Problem Concepts

Additional Information: Tips for Solving Age Problems

Age word problems are common in algebra and competitive exams. Here are some tips:

• Read Carefully: Understand the relationships between the ages of different people. Note down keywords like "older than," "younger than," "twice as old," "half as old," "in x years," "x years ago."
• Choose a Base Variable: Often, one person's age is described in terms of another's. Start by assigning a variable to the age of the person whose age is used as the base for comparison.
• Write Expressions: Translate each relationship into an algebraic expression using the variable(s) you defined.
• Formulate the Equation: Look for a condition involving the sum, difference, or total of the ages (possibly at a different time) and set up an equation.
• Solve the Equation: Use algebraic techniques to solve the equation for the variable.
• Check Your Answer: Plug the value(s) you found back into the original problem statement to ensure all conditions are met. Make sure you answer the specific question asked (e.g., find Goutam's age, not just Sai's age).
• Organize Information: Using a table can sometimes help keep track of ages at different points in time or the relationships between different people.