Shrink numbers within a range preserving distance ratio

I am not sure whether this problem would belong to Algebra or set Theory, hopefully it belongs here... I am not a mathematician, so its hard for me to come to a concrete category. I am trying to solve a problem at hand, and I need to get this necessary step done in order to find a complete solution.

Problem Description:

Suppose I have some points spread (unequally) on a line within some range. I want to shrink the points within a smaller range, so that the old distance ratio between points remain preserved.

e.g.

I have points a,b,c,d,e,f,g spread between Range X & Y

X = 0.25 Y = 0.90

a = 0.25 , b=0.32 , c=0.45, , d=0.50, e=0.60, f=0.80, , g=0.90

As we can see above that the distances are uneuqal between the spreaded points. Now, I want to shrink these points to a smaller range X' & Y',

where

X' = 0.35 Y' = 0.65

How would I calculate values of a',b',c',d',e',f',g' , such that the distance ration between all these points remain same for the smaller range.

a' = ? , b'=? , c'=?, , d'=?, e'=?, f'=?, , g'=?

I have been trying to come up with a formula that would help me solve this problem, but am unable to formulate one.

I hope someone here can help me.