What number should be added to each term of the ratio `2a : 3b ` so that the ratio of the additions will be `c : d`?

Explanation

Let the number x should be added to each term.
As per question, `(2a + x) : (3b + x) = c : d`
or, `(2a + x)/(3b + x) = c/d " or, 2ad " + dx = 3bc + cx`
or, ` dx - cx = 3bc -2ad`
or, ` x (d -c) = 3bc - 2ad`
or, ` x = (-(2ad - 3bc))/(d-c) " or , " x = (-(2ad - 3bc))/(-(c-d))`
or, ` x = (2ad - 3bc)/(c -d)`
Hence the required number ` = (2ad - 3bc)/(c-d)`