Understanding the Concept of Reversing Digits with a Two-Digit Number

Consider a two-digit number where the digits are in the ratio of 2:3. When 27 is added to this number, the digits are reversed. What is the original number?

Solution and Explanation

Let's denote the digits of the number as x (tens place) and y (units place). The ratio of the digits is 2:3, thus we can write:

Step 1: Establish the Ratio

x/y 2/3

To express x and y in terms of a single variable, we can write:

x 2k, y 3k

Step 2: Form the Equation

The original number can be written as 10y x. When 27 is added to this number, the digits are reversed, giving us:

10y x 27 1 y

Rearranging the terms, we get:

9y - 9x -27

Step 3: Simplify the Equation

We can further simplify the equation by substituting x and y:

9(3k) - 9(2k) -27

This simplifies to:

-9k -27

Solving for k:

k 3

Step 4: Find the Values of x and y

Substituting k 3 back into the expressions for x and y yields:

x 2(3) 6

y 3(3) 9

Conclusion

The original number is therefore 69.

Verification

Let's verify the solution by adding 27 to 69:

Verification Step 1: Addition

69 27 96

Indeed, the digits are reversed when 27 is added to 69, confirming our solution.

Alternative Solution Method

Method 1: Trial and Error

We identify all two-digit numbers where the digits are in the ratio of 2:3. These numbers are 23, 46, and 69. Only 69 meets the requirement that when 27 is added, the digits are reversed:

Verification Step 2: Adding 27 to 69

69 27 96 (Digits are reversed)

Method 2: Mathematical Representation

Let the number be xy, where x/y 2/3. This can be represented as:

3x - 2y 0

Further manipulation:

27 10y x 1 y

Rearrange to find:

9y - 9x -27

This simplifies to:

y - x 3

Resolving the system of equations:

3x - 2y 0

y - x 3

Solving these, we get:

x 6

y 9

Thus, the number is 69.

Additional Insights

The problem can also be solved by recognizing that the reverse of the digits increases the number by 27. The digit in the tens place is 27 divided by 9, which is 3 less than the digit in the units place. Given the sum of the digits is 9, the possible digits are 3 and 6, forming the number 36. However, 36 does not meet the criteria. The correct answer, as previously identified, is 69.

The final solution is the number 69, as it satisfies all the given conditions in the problem.