Solving the Age Word Problem

This problem asks us to find the age of Reena's father given the sum and difference of their ages. We can solve this using a system of linear equations.

Setting up Equations for Ages

Let's represent the ages with variables:

• Let \(R\) be Reena's age.
• Let \(F\) be her father's age.

According to the problem statement, we have two pieces of information:

1. The sum of Reena's and her father's age is 60. This can be written as an equation: $$R + F = 60 \quad \text{(Equation 1)}$$
2. The difference between their ages is 36. Since a father is typically older than their daughter, the difference is likely Father's age minus Reena's age. $$F - R = 36 \quad \text{(Equation 2)}$$

Solving the System of Linear Equations

We have a system of two linear equations with two variables:

Equation 1: \(R + F = 60\)

Equation 2: \(F - R = 36\)

We can solve this system using the elimination method. Notice that the \(R\) terms have opposite signs in the two equations (\(+R\) in Equation 1 and \(-R\) in Equation 2). If we add Equation 1 and Equation 2, the \(R\) terms will cancel out.

Add (Equation 1) and (Equation 2):

\((R + F) + (F - R) = 60 + 36\)

\(R + F + F - R = 96\)

\((R - R) + (F + F) = 96\)

\(0 + 2F = 96\)

\(2F = 96\)

Now, to find the father's age (\(F\)), divide both sides of the equation by 2:

\(\frac{2F}{2} = \frac{96}{2}\)

\(F = 48\)

Verification of the Ages

We found the father's age \(F = 48\) years. Now we can find Reena's age \(R\) using either Equation 1 or Equation 2.

Using Equation 1: \(R + F = 60\)

\(R + 48 = 60\)

\(R = 60 - 48\)

\(R = 12\)

So, Reena's age is 12 years.

Let's check if these ages satisfy Equation 2:

\(F - R = 36\)

\(48 - 12 = 36\)

\(36 = 36\)

The ages satisfy both conditions given in the problem.

Final Answer on Father's Age

Based on our calculations, Reena's father's age is 48 years.

The sum of their ages: \(12 + 48 = 60\)

The difference in their ages: \(48 - 12 = 36\)

Both conditions are met.

Age Problem Revision Table

Additional Information on Solving Age Problems

Age problems are common types of word problems in algebra. They often involve relationships between people's ages at different points in time (present, past, or future).

Key strategies for solving age problems:

• Read Carefully: Understand whose age is being asked for and what information is given about the relationships between ages.
• Assign Variables: Choose variables to represent the unknown ages, usually at the present time.
• Form Equations: Write equations based on the relationships described. Pay attention to phrases like "sum," "difference," "ratio," "times as old," "in X years," "X years ago."
• "In X years": If someone's current age is \(A\), their age in \(X\) years will be \(A + X\).
• "X years ago": If someone's current age is \(A\), their age \(X\) years ago was \(A - X\).
• Solve the Equations: Use algebraic methods (substitution, elimination) to solve the system of equations.
• Check Your Answer: Plug the calculated ages back into the original word problem or equations to ensure they make sense and satisfy all conditions.

This specific problem is a simple example involving just the sum and difference of ages at the present time, leading to a straightforward system of two linear equations.