The number obtained by interchanging the two digits of $a$ two digit number is less than the original number by 27. If the difference between the two digits of the number is 3, what is the original number?

Step 1 (Formulate Algebraic Number): Let the original two-digit number be represented by $10x+y$, where $x$ is the tens digit and $y$ is the units digit ($1\le x\le 9$ and $0\le y\le 9$). The number with interchanged digits is $10y+x$.

Step 2 (Set up Equations): We are given:

1. The interchanged number is less than the original by 27:

2. The difference between the two digits is 3:

Since the original number is larger than the interchanged one, we must have $x>y$, so $x-y=3$. This is identical to Equation 1.

Step 3 (Identify Solutions): Since the two equations are identical, any two-digit number with tens digit 3 more than its units digit works. The candidates are:

• $96$ (since $9-6=3$, interchanged is 69, $96-69=27$)

• $85,74,63,52,41,30$

Step 4 (Conclusion): A standard representative correct answer is 96.