Solving Father Son Age Problem

The question asks us to find the son's age after 6 years, given the sum of the present ages of the father and son, and a relationship between their ages six years ago.

Setting Up the Equations for Ages

Let's use variables to represent their current ages:

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

From the question, we have two pieces of information that can be translated into equations:

1. The sum of their present ages is 60 years.
   This gives us the equation: \(F + S = 60\) (Equation 1)
2. Six years ago, father’s age was five times the age of the son.
   Six years ago, the father's age was \(F - 6\).
   Six years ago, the son's age was \(S - 6\).
   The relationship is: \(F - 6 = 5 \times (S - 6)\) (Equation 2)

Solving the System of Equations

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

\(F + S = 60\)

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

We can solve this system using substitution. From Equation 1, we can express \(F\) in terms of \(S\):

\(F = 60 - S\)

Now, substitute this expression for \(F\) into Equation 2:

\((60 - S) - 6 = 5(S - 6)\)

Simplify the left side:

\(54 - S = 5(S - 6)\)

Distribute the 5 on the right side:

\(54 - S = 5S - 30\)

Now, we need to isolate \(S\). Let's move all terms with \(S\) to one side and constant terms to the other side. Add \(S\) to both sides and add 30 to both sides:

\(54 + 30 = 5S + S\)

\(84 = 6S\)

Now, divide by 6 to find the value of \(S\):

\(S = \frac{84}{6}\)

\(S = 14\)

So, the present age of the son is 14 years.

Calculating Son's Age After 6 Years

The question asks for the son's age after 6 years.

Son's age after 6 years = Present age of son + 6 years

Son's age after 6 years = \(14 + 6\)

Son's age after 6 years = \(20\) years

Therefore, the son's age will be 20 years after 6 years.

Verification

Let's quickly check if these ages satisfy the original conditions.

Present ages: Son = 14, Father = 60 - 14 = 46.

Sum of present ages: \(46 + 14 = 60\). This matches the first condition.

Ages six years ago: Son = \(14 - 6 = 8\), Father = \(46 - 6 = 40\).

Was father's age five times the son's age six years ago? \(5 \times 8 = 40\). Yes, \(40 = 5 \times 8\). This matches the second condition.

The calculated present ages are correct, and the son's age after 6 years is 20.

Revision Table: Key Concepts in Age Problems

Additional Information: Age Problem Strategies

Age problems are a common type of word problem in algebra. Here are some general strategies:

• Always define your variables clearly, usually representing the present ages.
• Read the problem carefully to identify information about past, present, and future ages.
• Write down the relationships between ages as mathematical equations. Pay close attention to phrases like "sum," "difference," "times," "years ago," and "years from now."
• If there are two unknowns (like father's age and son's age), you will typically need two independent equations to solve for them.
• Solve the system of equations using methods like substitution or elimination.
• Once you find the present ages, use them to answer the specific question asked (which might be an age in the past or future).
• Verify your answer by plugging the calculated ages back into the original statements of the problem to ensure they hold true.