How Long Until a Father’s Age is Double His Son’s Age?

Often, when we consider the relationship between a father and his son, we think of their ages and how they evolve over time. A common question that arises is, given their current ages, after how many years will the father be twice as old as his son?

Let's explore this problem in detail. Suppose the current age of the father is 45 years, and the son’s age is 15 years. We want to find out after how many years x, the father's age will be double the son's age.

Setting Up the Equation

We start by denoting the current ages of the father and son.

Father's age: 45 years

Son's age: 15 years

Let x be the number of years from now when the father’s age will be double the son’s age.

Age Calculation in x Years

In x years:

Father's age: 45 x Son's age: 15 x

Solving the Equation

According to the problem, we need to find x such that:

45 x 2 (15 x)

Expanding and rearranging the equation:

45 x 30 2x

45 - 30 2x - x

15 x

Hence, in 15 years, the father's age will be double the son's age.

Verification

In 15 years:

Father's age: 45 15 60 Son's age: 15 15 30

Indeed, 60 is double 30. Therefore, the solution is correct.

Educational Value of Algebraic Equations

This problem demonstrates the application of simple algebraic equations to real-life situations. It also shows the step-by-step process of solving equations involving the variable x. By setting up the equation and solving it systematically, we can determine the exact point in time when one age will be twice another.

Proof of the Solution

To solidify our solution, let's evaluate the initial condition given in the premise. If x 15:

Father's age in 15 years: 45 15 60

Son's age in 15 years: 15 15 30

Indeed, 60 is twice 30. Therefore, our solution is verified.

Additional Scenarios

Often, similar problems are used to test students' understanding of algebraic concepts. Here are a couple more scenarios for practice:

Problem 1

5 years ago, the father’s age was 3 times the son’s age. If we denote the father's current age as F and the son's current age as S, then:

F - 5 3S - 5 F 3S - 10

After 7 years, the father’s age will be twice the son’s age:

F 7 2(S 7) F 2S 14 - 7 F 2S 7

We now have two equations:

F 3S - 10 F 2S 7

Solving this system, we find:

3S - 10 2S 7

3S - 2S 7 10

S 17

Substituting S 17 back into one of the equations:

F 3(17) - 10 51 - 10 41

Therefore, the present age of the father is 41 years, and the son is 17 years.

Problem 2

Another similar problem is:

f - 4 6s - 4 f 16 2s 16 f 6s - 24 4 f 6s - 20 6s - 20 16 2s 32 6s - 4 2s 32 6s - 2s 32 4 4s 36 s 9 (son's age) f 6(9) - 20 54 - 20 f 34 (father's age)

Hence, the son is 9 years old, and the father is 34 years old.

Conclusion

Understanding how to set up and solve these types of equations is crucial for mastering algebra and enhancing problem-solving skills. Such exercises help to build a strong foundation in mathematics and are often tested in various academic and professional settings.