Understanding the Father Son Age Problem

This problem involves finding the current ages of a father and his son based on two pieces of information: the sum of their present ages and the ratio of their ages from a past time.

We are given:

• The sum of their present ages is 45 years.
• 5 years ago, the ratio of their ages was 6 : 1.

Our goal is to find the current age of the father.

Setting Up Equations for Ages

Let's use variables to represent the current ages:

• Let the current age of the father be \( F \) years.
• Let the current age of the son be \( S \) years.

From the first piece of information, the sum of their present ages is 45 years. This gives us our first equation:

\( F + S = 45 \)

Now, let's consider their ages 5 years ago:

• Father's age 5 years ago was \( F - 5 \) years.
• Son's age 5 years ago was \( S - 5 \) years.

From the second piece of information, the ratio of their ages 5 years ago was 6 : 1. This gives us our second equation:

\( \frac{F - 5}{S - 5} = \frac{6}{1} \)

Solving the System of Equations to Find Ages

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

1. \( F + S = 45 \)
2. \( \frac{F - 5}{S - 5} = 6 \)

From equation (1), we can express \( S \) in terms of \( F \):

\( S = 45 - F \)

Now substitute this expression for \( S \) into equation (2):

\( \frac{F - 5}{(45 - F) - 5} = 6 \)

Simplify the denominator:

\( \frac{F - 5}{40 - F} = 6 \)

Multiply both sides by \( (40 - F) \) to remove the denominator:

\( F - 5 = 6 \times (40 - F) \)

\( F - 5 = 240 - 6F \)

Now, we need to isolate \( F \). Add \( 6F \) to both sides of the equation:

\( F + 6F - 5 = 240 \)

\( 7F - 5 = 240 \)

Add 5 to both sides:

\( 7F = 240 + 5 \)

\( 7F = 245 \)

Divide by 7 to find the value of \( F \):

\( F = \frac{245}{7} \)

\( F = 35 \)

So, the current age of the father is 35 years.

Verifying the Age Solution

To check our answer, we can find the son's current age using \( S = 45 - F \):

\( S = 45 - 35 = 10 \)

The son's current age is 10 years.

Now, let's check their ages 5 years ago:

• Father's age 5 years ago: \( 35 - 5 = 30 \) years.
• Son's age 5 years ago: \( 10 - 5 = 5 \) years.

The ratio of their ages 5 years ago was \( 30 : 5 \). Dividing both by 5, we get \( 6 : 1 \). This matches the ratio given in the problem, confirming our calculated ages are correct.

The current age of the father is 35 years.

Revision Table: Key Concepts in Age Problems

Additional Information on Solving Age Word Problems

Age problems are a common type of word problem in algebra. They typically involve finding the present ages of one or more people based on conditions related to their ages at different points in time (past, present, or future).

Here are some tips for solving age problems:

• Read Carefully: Understand what ages (present, past, or future) and what relationships (sum, difference, ratio, product) are given.
• Assign Variables: Use variables (like \( x \), \( y \), \( F \), \( S \)) to represent the unknown ages, usually the present ages.
• Write Expressions for Ages at Different Times: If the problem refers to ages in the past or future, write expressions for those ages in terms of your variables. For example, if the present age is \( x \):
  • Age \( n \) years ago: \( x - n \)
  • Age \( n \) years from now: \( x + n \)
• Formulate Equations: Translate the given conditions about the sums, differences, ratios, or products of ages into algebraic equations.
• Solve the Equations: Use algebraic methods (like substitution, elimination) to solve the system of equations for the unknown variables.
• Check Your Answer: Plug the calculated ages back into the original problem statements to ensure they satisfy all conditions.
• Answer the Specific Question: Make sure you provide the answer requested in the question (e.g., father's age, son's age, sum of ages 10 years from now).