Solving the Father-Son Age Ratio Problem

This problem involves finding the current ages of a father and son based on given conditions and then determining the ratio of their ages in the future.

Let's break down the problem and solve it step-by-step.

Defining Variables for Ages

We start by assigning variables to represent the current ages:

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

Setting Up Equations from the Given Information

The problem provides two main pieces of information, which we can translate into mathematical equations:

1. The sum of their present ages is 52 years. $$F + S = 52 \quad \text{(Equation 1)}$$
2. Four years from now, the son's age will be \(\frac{1}{4}\) that of the father's age.
   • Four years from now, the father's age will be \(F + 4\).
   • Four years from now, the son's age will be \(S + 4\).
   The relationship is: $$S + 4 = \frac{1}{4}(F + 4) \quad \text{(Equation 2)}$$

Solving for Present Ages

Now, we solve the system of equations to find the values of \(F\) and \(S\).

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

Substitute this expression for \(F\) into Equation 2:

Multiply both sides by 4 to eliminate the fraction:

Collect terms with \(S\) on one side and constants on the other:

Solve for \(S\):

So, the present age of the son is 8 years.

Now, substitute the value of \(S\) back into Equation 1 (\(F = 52 - S\)) to find the father's present age:

The present age of the father is 44 years.

Calculating Ages 10 Years from Now

We need to find the ratio of their ages 10 years from now.

• Father's age 10 years from now will be \(F + 10\). $$44 + 10 = 54 \text{ years}$$
• Son's age 10 years from now will be \(S + 10\). $$8 + 10 = 18 \text{ years}$$

Determining the Ratio of Ages

The question asks for the ratio of the age of the son and father 10 years from now. This means the ratio is Son's Age : Father's Age.

Ratio = \((S + 10) : (F + 10)\)

Ratio = \(18 : 54\)

To simplify the ratio, find the greatest common divisor (GCD) of 18 and 54. The GCD is 18.

Divide both parts of the ratio by 18:

The ratio of the age of the son and father 10 years from now is \(1 : 3\).

Summary of Ages

Check the condition for 4 years hence: Son's age (12) is \(1/4\) of Father's age (48), since \(12 = \frac{1}{4} \times 48\). This matches the problem statement.

The ratio of son's age to father's age in 10 years is \(18 : 54\), which simplifies to \(1 : 3\).

Revision Table: Key Concepts

Additional Information: Age Problems and Ratios

Age-based problems are common in algebra. They often involve setting up equations based on relationships between ages at different points in time (past, present, future). Here are some tips for solving them:

• Always define your variables clearly, usually representing present ages.
• Carefully read the problem to understand if the age relationship is for the present, past, or future.
• When dealing with past or future ages, remember to add or subtract the same number of years from everyone's age.
• Ratios can be simplified by dividing both parts by their greatest common divisor.

Understanding how to translate word problems into equations is a fundamental skill. Practice with different scenarios involving sums, differences, products, and ratios of ages.