Question

The length of three line segments are a,b, and c respectively,
and a=2/3b c=3/2b then a : c = ?

Answer 1

a=2/3b
3a = 2b
b = 3a/2

c=3/2b
2c = 3b
b = 2c/3

thus: 2c/3 = 3a/2
4c = 9a
4/9 = a/c

a : c = 4 : 9

notice 2/3b ≠ 2/(3b)
I assumed you meant it the way you typed it.

Answer 2

a:b = 2:3
c:b = 3:2

we want b in the middle, so
b:c = 2:3

lcm(2,3) = 6, so now we can write

a:b = 4:6
b:c = 6:9

a:b:c = 4:6:9

a:c = 4:9

Answer 3

can you explain to me what you mean when you said you need "b" in the middle? thanks

Answer 4

b is bigger than a and smaller than c

Answer 5

it's not that b is the middle value, but that if we want to build a set of ratios,

a:b:c

we need to have a common value for b in the two ratios

a:b and b:c

We could not have placed a or c in the center spot, because we only had one ratio for each of those. But to join the two ratios, we need a single value for b.