Problem: Finding the Present Ages of P and Q Using Ratio and Age Calculations

In a certain scenario, the ratio of the present age of P and Q is given as 3:4. Five years ago, the ratio of their ages was 5:7. Our task is to determine the present ages of P and Q.

Step-by-Step Solution

We start by letting the present age of P be 3x years and the present age of Q be 4x years, since their current ages are in the ratio 3:4.

Step 1: Setting up the Equation

Five years ago, the ages of P and Q were 3x - 5 and 4x - 5, respectively. The ratio of their ages five years ago was given as 5:7. Therefore, we can set up the proportion:

3x - 5/4x - 5 5/7

Step 2: Cross-Multiplying and Simplifying

By cross-multiplying, we get:

7(3x - 5) 5(4x - 5)

Expanding both sides:

21x - 35 2 - 25

Transposing terms to isolate x:

21x - 2 35 - 25

x 10

Step 3: Calculating the Present Ages

Using the value of x we found, we can calculate the present ages of P and Q:

P’s present age 3x 3(10) 30 years

Q’s present age 4x 4(10) 40 years

Verification

For verification, we can check if the conditions given in the problem are satisfied:

Five years ago, the ages were:

P: 30 - 5 25 years

Q: 40 - 5 35 years

The ratio of these ages is:

25:35 5:7, which matches the given condition.

Conclusion

The present ages of P and Q are 30 and 40 years respectively. The use of algebraic equations and ratio analysis provides a systematic and accurate solution to the problem.

Further Learning

Understanding how to solve such problems is crucial in various fields, including data analysis and financial planning. Practicing similar problems can help in mastering algebra and improving logical reasoning skills.