This question is an example of an age word problem, which can be solved using linear equations. We are given information about the current ages of a father and daughter and their ages ten years ago. We need to find the father's present age.

Setting Up Equations for the Age Problem

Let's represent the unknown ages with variables:

• Let \( F \) be the present age of the father.
• Let \( D \) be the present age of the daughter.

We are given two conditions. Let's translate them into mathematical equations:

Condition 1: The age of a father is thrice the age of his daughter.

This refers to their present ages. So, we can write:

\[ F = 3D \quad \text{(Equation 1)} \]

Condition 2: Ten years ago, his age was five times his daughter's age.

First, let's find their ages ten years ago:

• Father's age ten years ago: \( F - 10 \)
• Daughter's age ten years ago: \( D - 10 \)

According to the condition, the father's age ten years ago was five times the daughter's age ten years ago:

\[ F - 10 = 5 \times (D - 10) \quad \text{(Equation 2)} \]

Solving the System of Equations

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

1. \( F = 3D \)
2. \( F - 10 = 5(D - 10) \)

We can solve this system using the substitution method. Since Equation 1 already gives us \( F \) in terms of \( D \), we can substitute the expression for \( F \) from Equation 1 into Equation 2.

Substitute \( F = 3D \) into Equation 2:

\[ (3D) - 10 = 5(D - 10) \]

Now, we need to solve this equation for \( D \). First, distribute the 5 on the right side:

\[ 3D - 10 = 5D - 50 \]

Next, gather the terms with \( D \) on one side and the constant terms on the other side. Let's move \( 3D \) to the right side and \( -50 \) to the left side:

\[ -10 + 50 = 5D - 3D \]

\[ 40 = 2D \]

Now, isolate \( D \) by dividing both sides by 2:

\[ D = \frac{40}{2} \]

\[ D = 20 \]

So, the present age of the daughter is 20 years.

The question asks for the present age of the father. We can find this using Equation 1, \( F = 3D \).

Substitute the value of \( D = 20 \) into Equation 1:

\[ F = 3 \times 20 \]

\[ F = 60 \]

Thus, the present age of the father is 60 years.

Verification

Let's check if these ages satisfy both conditions:

• Present: Father's age = 60, Daughter's age = 20. Is \( 60 = 3 \times 20 \)? Yes, \( 60 = 60 \). Condition 1 is satisfied.
• Ten years ago: Father's age = \( 60 - 10 = 50 \). Daughter's age = \( 20 - 10 = 10 \). Is \( 50 = 5 \times 10 \)? Yes, \( 50 = 50 \). Condition 2 is satisfied.

Both conditions are met, confirming our solution is correct.

The present age of the father is 60 years.

Revision Table: Age Word Problem Concepts

Additional Information: Solving Age Word Problems

Age word problems are common in algebra. They usually involve setting up equations based on relationships between people's ages at different points in time (present, past, or future).

Key steps to solve age word problems:

1. Identify Variables: Assign variables (like \( x, y \)) to the unknown present ages of the people involved.
2. Translate Conditions into Equations: Read the problem carefully and write down algebraic equations based on the given relationships between ages at different times. Remember to adjust ages for past or future scenarios (subtract for past, add for future).
3. Solve the System of Equations: Use methods like substitution or elimination to find the values of the variables.
4. Answer the Question: Make sure you answer what the question asks for (e.g., present age, age in the future, difference in ages).
5. Verify Your Solution: Plug the calculated ages back into the original word problem conditions to ensure they hold true. This helps catch any errors.

Practice with different types of age relationships (sum, difference, ratios, multiples) and time periods (past, future) to become proficient.