In an AP the difference between the 8th and the 4th term is 20 and the 8th term is 1½ times the fourth term.what is the common difference

Let the first term of the AP be $a$ and the common difference be $d$.
The $n$-th term of an AP is given by $a_n = a + (n-1)d$.

From the first condition: "the difference between the 8th and the 4th term is 20"
$a_8 - a_4 = 20$
$(a + (8-1)d) - (a + (4-1)d) = 20$
$(a + 7d) - (a + 3d) = 20$
$a + 7d - a - 3d = 20$
$4d = 20$ (Equation 1)

From the second condition: "the 8th term is 1½ times the fourth term"
$a_8 = 1\frac{1}{2} \times a_4$
$a_8 = \frac{3}{2} a_4$
$a + 7d = \frac{3}{2} (a + 3d)$ (Equation 2)

$4d = 20$
$d = \frac{20}{4}$
$d = 5$

Substitute $d=5$ into Equation 2:
$a + 7(5) = \frac{3}{2} (a + 3(5))$
$a + 35 = \frac{3}{2} (a + 15)$
Multiply both sides by 2 to clear the fraction:
$2(a + 35) = 3(a + 15)$
$2a + 70 = 3a + 45$
Subtract $2a$ from both sides:
$70 = a + 45$
Subtract 45 from both sides:
$a = 70 - 45$
$a = 25$

The common difference is 5.