we will assume that the number we are looking for has 2 digits

let

x = the ones digits

y = the tens digits

the number can be written as: x + 10y

the number where the digits are reversed is: y + 10x

ratio of tens digit to ones digit is 2:3

y : x = 2 : 3 cross-multiply

2x = 3y

if 27 is added to the number the digits are reversed:

x + 10y + 27 = y + 10x

9x - 9y = 27 divide both sides by 9

x - y = 3

by solving the system of equations:

2x = 3y

x - y = 3

we find:

x = 9

y = 6

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9 + 10*6 = 69

indeed 69 + 27 = 96

the number is 69.