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?6348966966

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

Step 2: The number obtained by interchanging the digits is 10y + x. According to the problem, this number is bigger than the original number by 27. So, we can write 10y + x = 10x + y + 27.

Step 3: Simplify the equation from step 2 to get 9y - 9x = 27. Divide the entire equation by 9 to get y - x = 3.

Step 4: 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 5: Multiply the first equation by y and the second equation by x to get: xy = 2/3y^2 and xy = x^2 + 3x.

Step 6: Set these two equations equal to each other to get: 2/3y^2 = x^2 + 3x.

Step 7: Solve this equation for y to get: y = sqrt((3x^2 + 9x) * 3/2).

Step 8: Substitute this value of y into the second equation from step 3 to get: sqrt((3x^2 + 9x) * 3/2) - x = 3.

Step 9: Solve this equation for x to get: x = 4.

Step 10: Substitute x = 4 into the first equation from step 1 to get: y = 6.

So, the original number is 46.