Understanding the Age Word Problem

This question is an age-based word problem that requires us to set up equations based on the information given about the ages of three individuals, T, P, and R, at different points in time. The goal is to find the present age of T.

Breaking Down the Problem Statements

Let's denote the present ages of T, P, and R as \(T_0\), \(P_0\), and \(R_0\) respectively. We can translate each statement into an algebraic equation:

• "Two years ago, T was twice as old as P."
• Two years ago, T's age was \(T_0 - 2\).
• Two years ago, P's age was \(P_0 - 2\).
• The relationship is \(T_0 - 2 = 2 \times (P_0 - 2)\).
• "P is thrice as old as R."
• This refers to their present ages.
• The relationship is \(P_0 = 3 \times R_0\).
• "In five years, P will be 29."
• In five years, P's age will be \(P_0 + 5\).
• The relationship is \(P_0 + 5 = 29\).

Solving for the Present Ages

We have a system of three equations:

1. \(T_0 - 2 = 2(P_0 - 2)\)
2. \(P_0 = 3R_0\)
3. \(P_0 + 5 = 29\)

We can start by solving the equation that involves only one unknown, which is the third equation involving P's future age.

Step 1: Find the Present Age of P (\(P_0\))

From the third equation:

\(P_0 + 5 = 29\)

Subtract 5 from both sides to isolate \(P_0\):

\(P_0 = 29 - 5\)

\(P_0 = 24\)

So, the present age of P is 24 years.

Step 2: Find the Present Age of T (\(T_0\))

Now that we know \(P_0 = 24\), we can use the first equation to find the present age of T:

\(T_0 - 2 = 2(P_0 - 2)\)

Substitute the value of \(P_0\):

\(T_0 - 2 = 2(24 - 2)\)

\(T_0 - 2 = 2(22)\)

\(T_0 - 2 = 44\)

Add 2 to both sides to isolate \(T_0\):

\(T_0 = 44 + 2\)

\(T_0 = 46\)

The present age of T is 46 years.

Step 3: (Optional) Find the Present Age of R (\(R_0\))

Although the question doesn't ask for R's age, we can find it using the second equation:

\(P_0 = 3R_0\)

Substitute \(P_0 = 24\):

\(24 = 3R_0\)

Divide by 3:

\(R_0 = \frac{24}{3}\)

\(R_0 = 8\)

The present age of R is 8 years.

Verifying the Solution

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

• Present ages: T = 46, P = 24, R = 8.
• "Two years ago, T was twice as old as P."
• Two years ago, T's age was \(46 - 2 = 44\).
• Two years ago, P's age was \(24 - 2 = 22\).
• Is \(44 = 2 \times 22\)? Yes, it is.
• "P is thrice as old as R."
• Present age of P is 24.
• Present age of R is 8.
• Is \(24 = 3 \times 8\)? Yes, it is.
• "In five years, P will be 29."
• In five years, P's age will be \(24 + 5 = 29\).
• Is \(24 + 5 = 29\)? Yes, it is.

All conditions are met. The present age of T is 46 years.

Age Problem Solution Summary

By setting up equations from the given statements and solving them step-by-step, we found the present age of T.

Revision Table: Key Concepts for Age Problems

Additional Information: Solving Age Word Problems

Age word problems are common in mathematics and competitive exams. They test your ability to translate relationships described in words into algebraic equations and solve them. Here are some general tips:

• Always start by defining variables for the present ages of the people involved. This makes setting up equations easier.
• Carefully read the time references. "Ago" means subtract from the present age; "in X years" means add to the present age.
• Look for relationships between the ages (e.g., one person is twice as old as another, the sum of their ages is a certain number).
• Write down each relationship as a separate equation.
• Solve the system of equations. Often, you can solve for one variable first and then substitute that value into other equations.
• Always verify your answer by checking if the calculated ages satisfy all the conditions given in the original problem.
• Practice with different types of age problems to become comfortable with various scenarios.