The digits of a two-digit number are in the ratio of 2 : 3 and the number obtained by interchanging the digits is bigger than the original number by 27.What is the original number?

Let's solve this step by step:

Step 1: Let's assume the original two-digit number is 10x + y, where x is the tens digit and y is the units digit.

Step 2: According to the problem, the digits x and y are in the ratio 2:3. So, we can write x/y = 2/3.

Step 3: The number obtained by interchanging the digits is bigger than the original number by 27. So, the interchanged number is 10y + x. According to the problem, 10y + x = 10x + y + 27.

Step 4: Simplify the equation from step 3 to get 9y - 9x = 27. Divide the entire equation by 9 to get y - x = 3.

Step 5: Now we have a system of two equations: x/y = 2/3 and y - x = 3. We can solve this system of equations to find the values of x and y.

Step 6: Multiply the first equation by 3 and the second equation by 2 to make the coefficients of y the same in both equations. We get 3x = 2y and 2y - 2x = 6.

Step 7: Subtract the second equation from the first to get x = 6. Substitute x = 6 into the first equation to get y = 9.

Step 8: Therefore, the original number is 10x + y = 10*6 + 9 = 69.