Let's solve the problem step by step.

Let the original two-digit number be represented as $10x + y$, where $x$ is the tens digit and $y$ is the units digit.

Step 1: Equation for the sum of the digits

We are told that the sum of the digits is 15: $x + y = 15 \quad \text{(Equation 1)}$

Step 2: Equation for the reversed number

When the digits are reversed, the new number becomes $10y + x$. We are also told that the new number is 27 more than the original number: $10y + x = (10x + y) + 27 \quad \text{(Equation 2)}$

Step 3: Simplifying Equation 2

Simplify the equation: $10y + x = 10x + y + 27$ Now, subtract $x$ and $y$ from both sides: $9y - 9x = 27$ Divide through by 9: $y - x = 3 \quad \text{(Equation 3)}$

Step 4: Solve the system of equations

Now, solve the system of equations:

1. $x + y = 15$
2. $y - x = 3$

From Equation 3, we have $y = x + 3$. Substitute this into Equation 1: $x + (x + 3) = 15$ Simplify: $2x + 3 = 15$ Subtract 3 from both sides: $2x = 12$ Divide by 2: $x = 6$

Step 5: Find $y$

Now substitute $x = 6$ into Equation 1: $6 + y = 15$ Thus, $y = 9$.

Step 6: Find the original number

The original number is $10x + y = 10(6) + 9 = 69$.

Conclusion

The original number is 69.

Would you like more details or have any questions? Here are some additional related questions:

1. What happens if the sum of the digits were 16 instead of 15?
2. How do we solve if the reversed number was 30 more than the original number?
3. What if the difference between the digits was given instead of the sum?
4. Can you express the solution process in terms of matrices?
5. How would the solution change if the digits represented a three-digit number?

Tip: When solving word problems, translating words into mathematical expressions step by step ensures clarity and accuracy.