Divide 96 into four parts which are in A.P. and the ratio between product of their means to product of their extremes is 15 : 7.

Let the four terms of A.P. are

(a - 3d), (a - d), (a + d), (a + 3d).

According to question,

⇒ a - 3d + a - d + a + d + a + 3d = 96

⇒ 4a = 96

⇒ a = 24.

So, terms are,

24 - 3d, 24 - d, 24 + d, 24 + 3d.

Given, ratio between product of their means to product of their extremes is 15 : 7.

$\therefore \dfrac{(24 - d)(24 + d)}{(24 - 3d)(24 + 3d)} = \dfrac{15}{7} \\[1em] \Rightarrow \dfrac{576 - d^2}{576 - 9d^2} = \dfrac{15}{7} \\[1em] \Rightarrow 7(576 - d^2) = 15(576 - 9d^2) \\[1em] \Rightarrow 4032 - 7d^2 = 8640 - 135d^2 \\[1em] \Rightarrow 135d^2 - 7d^2 = 8640 - 4032 \\[1em] \Rightarrow 128d^2 = 4608 \\[1em] \Rightarrow d^2 = 36 \\[1em] \Rightarrow d = \pm 6.$

Let d = 6,

Terms = (a - 3d), (a - d), (a + d), (a + 3d) = (24 - 3 × 6), (24 - 6), (24 + 6), (24 + 3 × 6)

= 6, 18, 30, 42.

Let d = -6,

Terms = (a - 3d), (a - d), (a + d), (a + 3d) = (24 - 3 × -6), (24 - (-6)), (24 + (-6)), (24 + 3 × -6)

= 42, 30, 18, 6.

Hence, four parts of 96 are 6, 18, 30, 42 or 42, 30, 18, 6.