CHRIST MATH
posted by pavithra .
Divide 20 into 4 parts which are in AP such that ratio between the product of the 1st part and the 4th part to the 2nd and 3rd part is 2:3 find the AP

What does AP mean and what does this have to do with Christ?

I will assume that AP stands for arithmetic progression
so you are given:
a + (a+d) + (a+2d) + (a+3d) = 20
4a + 6d = 20
2a + 3d = 10
product of 1st and 4th = a(a+3d)
product of 2nd to 3rd = (a+d)(a+2d)
then:
a(a+3d)/( (a+d)(a+2d) ) = 2/3
(a^2 + 3ad)/(a^2 + 3ad + 2d^2) = 2/3
3a^2 + 9ad = 2a^2 + 6ad + 4d^2 = 0
a^2 + 3ad  4d^2 = 0
(ad)(a+4d) = 0
a = d or a = 4d
case1: a = d
in 2a+3d=10
5d = 10,
d = 2, so a=d = 2
the terms are 2, 4, 6 , 8
case2: a = 4d
in 2a + 3d = 10
4d + 3d = 10
d = 10
d = 10, then a = 40
the terms are 40, 30, 20, 10 
I just noticed a typo in my second case
It should say:
case2: a = 4d
in 2a + 3d = 10
8d + 3d = 10
2d = 10
d = 5, then a = 20
the terms are 20 , 15, 10 , 5
but these do not add up to 20 , even though the ratio of product property works
so the only solution are the terms
2, 4, 6, 8
notice they add up to 20
and (2x8)/(4x6) = 16/24 = 2/3