The present ages of two students …

Given the ratio of the ages of two students = 5 : 3
Let their ages be 5<m:omath><m:r>x</m:r></m:omath> and 3<m:omath><m:r>x</m:r></m:omath>
After 6 years, their ages will be 5<m:omath><m:r>x</m:r></m:omath> + 6 and 3<m:omath><m:r>x</m:r></m:omath> + 6
Ratio of their ages after 6 years = 7 : 5
According to the question:
5<m:omath><m:r>x</m:r></m:omath> + 6 : 3<m:omath><m:r>x</m:r></m:omath> + 6 = 7 : 5
Product of mean = Product of extreme
⇒ 5(5<m:omath><m:r>x</m:r></m:omath> + 6) = 7 (3<m:omath><m:r>x</m:r></m:omath> + 6)
⇒ 25<m:omath><m:r>x</m:r></m:omath> + 30 = 21<m:omath><m:r>x</m:r></m:omath> + 42
⇒ 25<m:omath><m:r>x</m:r></m:omath> - 21<m:omath><m:r>x</m:r></m:omath> = 42 – 30
⇒ 4<m:omath><m:r>x</m:r></m:omath> = 8<m:omath><m:r>
⇒ x</m:r></m:omath> = 2
Hence the present ages of two students are 5(2) = 10 years and 3(2) = 6 years.