4points: a, b,c and d are marked in order on a line such that ab: ac = 3: 7 and bd: cd= 9: 2. if cd is 100cm long, find the length of ab.

You can create equations to find the lengths and how they compare.

The line is in order of A-B-C-D
Both ratios share BC so we can use that to transfer from one ratio to the next.
  
AB =3x  AC=7x   
subtract them to find the measurement of BC

AC- AB = BC
7x-3x = 4x .  BC = 4x

BD= 9y CD=2y

Subtract them to find the measurement of BC

BD - CD = BC

9y-2y =7y  BC =7y

CD is 100cm so   2y= 100 or y=50

substitute y for bc
7(50)=350 

substitute BC for terms of x

350 = 4x

divide both sides by 4 
X= 87.5

Substitute for AB
AB = 3x

AB = 3(87.5) = 262.5cm