This problem involves finding the present age of a son given information about the sum of his and his father's present ages and the ratio of their ages after a certain number of years.

Understanding the Father and Son Age Problem

We are given two main pieces of information:

1. The sum of the present ages of the father and the son is 60 years.
2. 5 years from now, the ratio of their ages will be 5 : 2.

Our goal is to determine the son's current age.

Setting Up Equations for Ages

Let's use variables to represent their current ages:

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

From the first piece of information, we can write the equation:

\(\qquad F + S = 60 \quad (Equation\ 1)\)

Now, let's consider their ages 5 years from now:

• Father's age 5 years from now will be \(F + 5\).
• Son's age 5 years from now will be \(S + 5\).

The second piece of information tells us the ratio of these future ages is 5 : 2. We can write this as:

\(\qquad \frac{F + 5}{S + 5} = \frac{5}{2} \quad (Equation\ 2)\)

Solving the System of Equations

We now have a system of two linear equations with two variables (\(F\) and \(S\)). We can solve this system to find the values of \(F\) and \(S\).

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

\(\qquad F = 60 - S\)

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

\(\qquad \frac{(60 - S) + 5}{S + 5} = \frac{5}{2}\)

Simplify the numerator:

\(\qquad \frac{65 - S}{S + 5} = \frac{5}{2}\)

To solve for \(S\), we can cross-multiply:

\(\qquad 2 \times (65 - S) = 5 \times (S + 5)\)

Distribute the numbers on both sides:

\(\qquad 130 - 2S = 5S + 25\)

Now, we need to gather the \(S\) terms on one side and the constant terms on the other side. Add \(2S\) to both sides:

\(\qquad 130 = 5S + 2S + 25\)

\(\qquad 130 = 7S + 25\)

Subtract 25 from both sides:

\(\qquad 130 - 25 = 7S\)

\(\qquad 105 = 7S\)

Finally, divide by 7 to find the value of \(S\):

\(\qquad S = \frac{105}{7}\)

\(\qquad S = 15\)

The Son's Current Age

The value we found for \(S\) is 15. Since \(S\) represents the present age of the son, the son's current age is 15 years.

Verification

Let's check if this answer fits the original conditions:

• If the son's current age is 15, and the sum of their ages is 60, then the father's current age is \(F = 60 - 15 = 45\) years.
• 5 years from now, the son's age will be \(15 + 5 = 20\) years.
• 5 years from now, the father's age will be \(45 + 5 = 50\) years.
• The ratio of their ages 5 years from now is \(\frac{50}{20} = \frac{5}{2}\), which matches the ratio given in the problem.

The calculated age satisfies both conditions, confirming that the son's current age is 15 years.

Revision Table: Key Concepts

Additional Information: Age Word Problems

Age-related word problems are common in algebra and quantitative aptitude tests. They typically involve relationships between people's ages at different points in time (past, present, future). The key to solving these problems is carefully setting up equations based on the given information for each time period mentioned.

• Representing Ages: If a person's current age is \(A\), their age \(x\) years ago was \(A - x\), and their age \(y\) years from now will be \(A + y\).
• Forming Relationships: Look for clues about sums, differences, products, or ratios of ages at different times.
• Using Multiple Variables: For problems involving two or more people, use a different variable for each person's current age.
• Consistent Time Frame: Ensure that when forming equations, you are comparing ages at the same point in time (e.g., comparing father's age 5 years from now to son's age 5 years from now).
• Solving Techniques: Systems of equations can be solved using substitution (as demonstrated above), elimination, or graphical methods. Substitution is often effective for age problems where one variable can easily be expressed in terms of another.

Practicing different types of age problems helps build confidence in translating verbal descriptions into mathematical models.