The problem at hand involves solving a ratio problem to find the present ages of two girls. The ages of the girls are in a current ratio of 5:7, and we are given an additional piece of information: eight years ago, their ages were in a ratio of 7:13. This problem requires the application of algebraic equations to find the present ages of both girls.

Solving the Problem Step-by-Step

Step 1: Define the Variables

Let's denote the present age of the two girls as 5x and 7x, respectively.

Current Ratio

The current ratio of their ages is given as 5:7. Therefore, if we let the age of the first girl be 5x and the second girl be 7x, we can write the current ratio as:

5x : 7x

Eight Years Ago

Eight years ago, their ages were 5x - 8 and 7x - 8, respectively. According to the problem, the ratio of their ages eight years ago was 7:13. Therefore, we can write the equation as:

5x - 8 7x - 8 7 13

By cross-multiplying the equation, we get:

13 ( 5x - 8 ) 7 ( 7x - 8 )

Simplifying the equation:

65x - 104 49x - 56

Rearranging the terms:

65x - 49x 104 - 56 → 16x 48 → x 3

Now, substituting x back into the original equations:

5x 5 ? 3 15

7x 7 ? 3 21

Thus, the present ages of the two girls are 15 years and 21 years, respectively.

Conclusion

We have successfully solved the problem of finding the ages of two girls using the given ratios. By setting up the correct algebraic equations and solving them step-by-step, we were able to determine their current ages. This process demonstrates the importance of logical reasoning and the application of mathematical principles to real-world problems.