Original Number Problem Overview

This problem asks us to find a specific two digit number based on two key pieces of information. We are told about the sum of the digits and how a new number formed by reversing the digits relates to the original number. To solve this, we will use algebraic equations, which is a common approach for number puzzles.

Digit Representation and Conditions

To accurately represent the original number and set up our equations, let's assign variables to its digits:

• Let \(x\) represent the tens digit of the original number.
• Let \(y\) represent the units digit of the original number.
• Based on these digits, the value of the original number can be written as \(10x + y\).

Now, let's translate the given conditions from the question into mathematical equations.

Condition 1: Sum of the Digits

The first statement in the question is: "The sum of the digits of a two digit number is 12."

• This directly translates into our first equation:
• \[x + y = 12 \quad \text{(Equation 1)}\]

Condition 2: Reversing the Digits

The second statement describes what happens when the digits are reversed: "If the new number formed by reversing the digits is greater than the original number by 54..."

• When the digits are reversed, \(y\) becomes the tens digit and \(x\) becomes the units digit.
• So, the new number formed by reversing the digits is \(10y + x\).
• The condition states that this new number is 54 more than the original number. We can write this as:
• \[(10y + x) - (10x + y) = 54\]
• Let's simplify this equation by combining like terms:
• \[10y - y + x - 10x = 54\]
• \[9y - 9x = 54\]
• We can simplify this equation further by dividing every term by 9:
• \[\frac{9y}{9} - \frac{9x}{9} = \frac{54}{9}\]
• \[y - x = 6 \quad \text{(Equation 2)}\]

Solving for the Digits

Now we have a system of two simple linear equations:

1. Equation 1: \(x + y = 12\)
2. Equation 2: \(y - x = 6\)

We can solve this system using the elimination method. If we add Equation 1 and Equation 2, the \(x\) terms will cancel out:

From the result \(2y = 18\), we can easily find the value of \(y\):

• \[y = \frac{18}{2}\]
• \[y = 9\]

Now that we have \(y = 9\), we can substitute this value back into either Equation 1 or Equation 2 to find \(x\). Let's use Equation 1 (\(x + y = 12\)):

• Substitute \(y = 9\):
• \[x + 9 = 12\]
• Subtract 9 from both sides to isolate \(x\):
• \[x = 12 - 9\]
• \[x = 3\]

Finding the Original Number

We have successfully found the values for both digits: the tens digit \(x = 3\) and the units digit \(y = 9\).

The original number was defined as \(10x + y\). Let's substitute the values we found:

• \[\text{Original Number} = 10(3) + 9\]
• \[\text{Original Number} = 30 + 9\]
• \[\text{Original Number} = 39\]

Verification of the Solution

It's always a good practice to check our answer against the original problem conditions to ensure accuracy. Our calculated original number is 39.

• Check Condition 1 (Sum of the digits): The digits of 39 are 3 and 9. Their sum is \(3 + 9 = 12\). This matches the first condition provided in the question.
• Check Condition 2 (Reversing the digits): If we reverse the digits of 39, the new number formed by reversing the digits is 93.
• Now, let's see if the new number is greater than the original number by 54: \(93 - 39 = 54\). This also perfectly matches the second condition in the question.

Since both conditions are satisfied, our determined original number of 39 is correct.