To solve this problem, we need to find the present ages of the two girls, given that their current ages are in the ratio 5:7 and their ages eight years ago were in the ratio 7:13.
### Step-by-Step Solution:
1. Define Variables:
- Let the present age of the younger girl be [tex]\(5x\)[/tex].
- Let the present age of the older girl be [tex]\(7x\)[/tex].
2. Express Ages Eight Years Ago:
- The age of the younger girl eight years ago would be [tex]\(5x - 8\)[/tex].
- The age of the older girl eight years ago would be [tex]\(7x - 8\)[/tex].
3. Set Up the Ratio Eight Years Ago:
- According to the problem, eight years ago, their ages were in the ratio 7:13.
- Therefore, we write the equation:
[tex]\[
\frac{5x - 8}{7x - 8} = \frac{7}{13}
\][/tex]
4. Cross Multiply to Solve for [tex]\(x\)[/tex]:
- Cross-multiplying the equation, we get:
[tex]\[
13(5x - 8) = 7(7x - 8)
\][/tex]
5. Simplify the Equation:
- Distribute on both sides:
[tex]\[
65x - 104 = 49x - 56
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
- Isolate [tex]\(x\)[/tex] by first moving all terms involving [tex]\(x\)[/tex] to one side:
[tex]\[
65x - 49x = 104 - 56
\][/tex]
[tex]\[
16x = 48
\][/tex]
[tex]\[
x = 3
\][/tex]
7. Find the Present Ages:
- Now that we have [tex]\(x = 3\)[/tex], we can calculate their present ages:
- The present age of the younger girl [tex]\(5x = 5 \times 3 = 15\)[/tex] years.
- The present age of the older girl [tex]\(7x = 7 \times 3 = 21\)[/tex] years.
### Conclusion:
The present ages of the girls are:
- The younger girl is 15 years old.
- The older girl is 21 years old.
