Solving a Complex Age Ratio Problem

The given problem involves complex algebraic equations and ratios, which might appear challenging at first glance. However, by breaking down the problem into manageable steps, we can find the solution. Let's dive into the detailed analysis and solution of this problem.

Introduction

We are given a problem about the age ratio of a father and a son. Currently, the ratio of the father's age to the son's age is 5:1. After 4 years, this ratio will change to 7:2. We need to find the sum of their current ages.

Step-by-Step Solution

To solve this problem, we will use algebraic equations. Let's denote the current age of the father as (x) and the son's age as (y).

First Condition

The current ratio of the ages of the father and the son is 5:1. This can be expressed as:

[ frac{x}{y} frac{5}{1} implies x 5y ]

Second Condition

After 4 years, the new ratio of their ages will be 7:2. This can be written as:

[ frac{x 4}{y 4} frac{7}{2} ]

Rearranging this equation, we get:

[ 2(x 4) 7(y 4) ]

[ 2x 8 7y 28 ]

[ 2x - 7y 20 ]

Solving for (x) and (y)

Using the first condition, substitute (x 5y) into the second condition:

[ 2(5y) - 7y 20 ]

[ 10y - 7y 20 ]

[ 3y 20 ]

[ y frac{20}{3} approx 6.67 text{ years} ]

Now, substituting (y) back into the first condition to find (x):

[ x 5y 5 times frac{20}{3} frac{100}{3} approx 33.33 text{ years} ]

The sum of their current ages is:

[ x y frac{100}{3} frac{20}{3} frac{120}{3} 40 text{ years} ]

Verification

Let's verify our solution by checking the ratio after 4 years:

After 4 years, their ages will be (x 4 frac{100}{3} 4 frac{100 12}{3} frac{112}{3}) and (y 4 frac{20}{3} 4 frac{20 12}{3} frac{32}{3}).

The new ratio is:

[ frac{frac{112}{3}}{frac{32}{3}} frac{112}{32} frac{28}{8} frac{7}{2} ]

This matches our second condition, confirming our solution is correct.

Conclusion

The sum of the current ages of the father and the son is 40 years.