Age Ratio Increase Over Time: A Mathematical Analysis

Have you ever wondered how the ratios of ages between two individuals will change over time? In this article, we will explore a specific case where the ratio of the ages of P and Q is 6:7, and we will dive into the calculations to determine their future age ratio after 4 years. This analysis not only helps us understand the concept but also provides valuable insights into how age ratios evolve.

Introduction to the Problem

The problem at hand involves two individuals, P and Q, with an initial age ratio of 6:7. According to the problem, Q is 4 years older than P. Our task is to determine the ratio of their ages after 4 years have passed.

Initial Conditions and Variables

Let's denote the present age of P as (6x) and the present age of Q as (7x). Given that Q is 4 years older than P, we can set up the following equation:

(7x 6x 4)

By solving for (x), we obtain:

(x 4)

Thus, the present ages of P and Q are:

Present age of P: (6x 6 times 4 24) years Present age of Q: (7x 7 times 4 28) years

Future Ages Calculation

After 4 years, we calculate their respective future ages:

Age of P after 4 years: (24 4 28) years Age of Q after 4 years: (28 4 32) years

Final Ratio Calculation

The final step is to find the ratio of their ages after 4 years:

(frac{28}{32} frac{7}{8})

Hence, the ratio of the ages of P and Q after 4 years will be 7:8.

Step-by-Step Analysis with Different Approaches

Approach 1: Direct Substitution

Let the present age of P be (6x) and the present age of Q be (7x). According to the problem, Q is 4 years older than P:

(7x 6x 4)

Solving for (x):

(x 4)

Thus, the present ages are:

Present age of P: (6x 24) years Present age of Q: (7x 28) years

After 4 years, their ages will be:

Age of P after 4 years: (24 4 28) years Age of Q after 4 years: (28 4 32) years

The ratio of their ages after 4 years is:

(frac{28}{32} frac{7}{8})

Approach 2: Using General Formulation

Given the ratio P:Q 6:7, let the age of P be (x) and the age of Q be (y). Since Q is 4 years older than P:

(y x 4)

The ratio equation gives:

(frac{x}{y} frac{6}{7})

Substituting (y x 4):

(frac{x}{x 4} frac{6}{7})

Solving for (x):

(7x 6(x 4))

(7x 6x 24)

(x 24)

Thus, (y 28)

After 4 years, their ages will be:

Age of P after 4 years: (24 4 28) years Age of Q after 4 years: (28 4 32) years

The ratio of their ages after 4 years is:

(frac{28}{32} frac{7}{8})

Conclusion

In conclusion, the mathematical analysis using various approaches confirms that the ratio of the ages of P and Q after 4 years will be 7:8. This problem illustrates the importance of setting up equations and solving them step-by-step to reach a definitive answer. Understanding age ratio calculations can be useful in various real-life scenarios and mathematical applications.