Solving Simultaneous Equations with Father and Son’s Age Problems

Understanding the relationship between the ages of a father and son can be a fun and practical application of mathematics. This article will explore a series of simultaneous equations involving these relationships, taking you through step-by-step solutions. Mastery of these problems not only enhances your algebraic skills but also provides insights into real-world mathematical concepts.

Simultaneous Equations in Word Problems

Let's start with a common type of problem: A father is 10 times the age that his son is now. In 6 years, the father will be 4 times the age of his son. How old is the son now?

Formulating the Equations

Let F represent the father's current age and S represent the son's current age. We have two key pieces of information:

1. F 10S

2. In 6 years, the father's age will be 4 times the son's age, so F 6 4(S 6)

Solving the Equations

Substitute the first equation into the second:

10S 6 4(S 6)

Simplify and solve for S:

10S 6 4S 24 10S - 4S 24 - 6 6S 18 S 3

The son is currently 3 years old. To find the father's age, use the first equation:

F 10S 10 × 3 30
So, the father is currently 30 years old.

In 6 years, the son will be 9 and the father will be 36, making the father 4 times the son's age.

More Simultaneous Equations

Let's explore another scenario:

5 years ago, the father's age was three times the son's age. In 7 years, the father's age will be twice the son's age. What are their present ages?

Formulating the Equations

Again, let F represent the father's current age and S represent the son's current age. We have:

1. 5 years ago, the father's age was 3 times the son's age, so F - 5 3(S - 5)

2. In 7 years, the father's age will be twice the son's age, so F 7 2(S 7)

Solving the Equations

First, simplify the equations:

F - 5 3S - 15
F 3S - 10

F 7 2S 14
F 2S 7

Equate the two expressions for F:

3S - 10 2S 7 3S - 2S 7 10 S 17

The son is currently 17 years old. To find the father's age, use the second equation:

F 2S 7 2 × 17 7 34 7 41
So, the father is currently 41 years old.

Additional Examples

Consider another situation where:

The son is 3 years old, and the father is 30, which is 10 times the son's age. In 6 years, the son will be 9, and the father will be 36, making the father 4 times the son's age.

Let S represent the son's current age and F represent the father's current age. We have:

1. F 3S

2. In 6 years, the father's age will be 4 times the son's age, so F 6 4(S 6)

Solving the Equations

Substitute the first equation into the second:

3S 6 4(S 6)

Simplify and solve for S:

3S 6 4S 24 3S - 4S 24 - 6 -S 18 S -18

Since age cannot be negative, re-examine the scenario and equations.

Try:

16 4X6 16 4X24 1 - 4X6 - 6 4X - 4X24 - 6 6X 18 X 3
So, the son is currently 3 years old. The father's age is 10 times the son's age, making the father 30 years old.

In 6 years, the son will be 9 and the father will be 36, making the father 4 times the son's age.

Another solution:

16 4X6 16 4X24 1 - 4X6 - 6 4X - 4X24 - 6 6X 18

X 3
So, the son is currently 3 years old, and the father is 30 years old in 6 years, making the father 4 times the son's age.

A more complex solution involves another set of equations:

Let 1S Y
16 4Y 6 - 10 6Y 10 3Y 10 7Y 10 40 4S 60 S 15 Y 10 40 2S 40 16 10 - 10 Therefore, the son is 15 years old and the father is 40 years old.

Conclusion

These simultaneous equations involving the age of a father and son are not only mathematically intriguing but also demonstrate practical applications of algebra. By solving these equations, you can explore age-related problems and gain a deeper understanding of mathematical concepts.