Understanding the 2-Digit Number Problem

This problem requires us to find a specific 2-digit number using information about the sum of its digits and the relationship between the original number and the number formed by reversing its digits.

Let's represent the 2-digit number. We can denote the digit in the tens place as '$a$' and the digit in the units place as '$b$'. Therefore, the value of the original number can be expressed algebraically as $10a + b$.

Formulating Algebraic Equations

The problem provides two key pieces of information that we can translate into equations:

• Condition 1: Sum of the digits is 12.

  This directly translates to the equation:

  $a + b = 12 \quad \text{(Equation 1)}$
• Condition 2: Interchanged digits number is 15 more than twice the original number.

  The number formed by interchanging the digits is $10b + a$. The original number is $10a + b$. The condition states:

  $10b + a = 2(10a + b) + 15$

  To simplify this, we first distribute the 2 on the right side:

  $10b + a = 20a + 2b + 15$

  Now, let's rearrange the terms to gather variables on one side and the constant on the other. We aim to get a standard form like $Ax + By = C$.

  $10b - 2b + a - 20a = 15$ $8b - 19a = 15$

  It's often helpful to write the variables in alphabetical order:

  $-19a + 8b = 15 \quad \text{(Equation 2)}$

Solving the System of Equations

We now have a system of two linear equations:

1. $a + b = 12$
2. $-19a + 8b = 15$

We can use the substitution method to solve for '$a$' and '$b$'. From Equation 1, we can express '$b$' in terms of '$a$':

Substitute this expression for '$b$' into Equation 2:

Distribute the 8:

Combine the '$a$' terms:

Isolate the term with '$a$' by subtracting 96 from both sides:

Solve for '$a$' by dividing both sides by -27:

Now, substitute the value $a = 3$ back into Equation 1 to find the value of '$b$':

Identifying the Original Number

We found the tens digit '$a$' to be 3 and the units digit '$b$' to be 9.

The original number is $10a + b$. Plugging in the values:

Original Number = $10(3) + 9 = 30 + 9 = 39$.

Verifying the Conditions

Let's check if the number 39 satisfies the conditions stated in the problem:

• Check Condition 1 (Sum of digits): The digits are 3 and 9. Their sum is $3 + 9 = 12$. This condition is met.
• Check Condition 2 (Interchanged number relation):
  • The original number is 39.
  • The number with digits interchanged is 93.
  • Twice the original number plus 15 is $2 \times 39 + 15 = 78 + 15 = 93$.
  The interchanged number (93) is equal to twice the original number plus 15 (93). This condition is also met.

Both conditions are satisfied for the number 39.

Evaluating the Options

The question asks for the original number, and our calculations based on the problem statement yield 39. The options provided were:

Our derived answer is 39, which corresponds to Option 1. The question indicates Option 4 (75) as the correct answer.