Define variables

Let the sum of the two students' present ages be $S$ and the teacher's age be $T$. Given $T = 1.5S$, we can write $T = \frac{3}{2}S$.

$T = \frac{3}{2}S$

Future age relationship

After 6 years, the teacher's age is $T+6$. Each of the two students also ages by 6 years, so the new sum of students' ages is $S + 6 + 6 = S + 12$.

$\frac{T + 6}{S + 12} = \frac{6}{5}$

Solve the system

Substitute $T = \frac{3}{2}S$ into the ratio equation: $5(1.5S + 6) = 6(S + 12)$. Expanding gives $7.5S + 30 = 6S + 72$, which simplifies to $1.5S = 42$, or $S = 28$.

$1.5S = 42 \implies S = 28$

Calculate Teacher's age

Using $T = 1.5S$, we find $T = 1.5 \times 28 = 42$.

$T = 42$