The digits of a two-digit number differ by 3. If the digits are interchanged and the resulting number is added to the original number, we get 143. What can be the original number?

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Points to Remember:

  • The problem involves a two-digit number.
  • The difference between the digits is 3.
  • Interchanging the digits and adding the result to the original number yields 143.
  • We need to find the original two-digit number.

Introduction:

This is a mathematical word problem requiring an analytical approach. We are presented with a scenario involving a two-digit number where the digits have a specific relationship. By applying algebraic principles and logical reasoning, we can determine the original number. This type of problem tests our understanding of place value and the manipulation of equations.

Body:

1. Defining Variables and Setting up Equations:

Let’s represent the tens digit of the original two-digit number as ‘x’ and the units digit as ‘y’. The original number can then be expressed as 10x + y. We are given that the difference between the digits is 3, which can be represented as:

  • x – y = 3 (or y – x = 3, depending on which digit is larger)

When the digits are interchanged, the new number is 10y + x. Adding this to the original number gives 143:

  • (10x + y) + (10y + x) = 143
  • 11x + 11y = 143
  • x + y = 13 (Dividing both sides by 11)

2. Solving the System of Equations:

Now we have a system of two linear equations:

  • x – y = 3
  • x + y = 13

We can solve this system using either substitution or elimination. Let’s use elimination: Adding the two equations together eliminates ‘y’:

  • 2x = 16
  • x = 8

Substituting x = 8 into either equation (let’s use x + y = 13):

  • 8 + y = 13
  • y = 5

Therefore, the original number is 85.

3. Verification:

Let’s check our solution:

  • Original number: 85
  • Digits differ by 3 (8 – 5 = 3)
  • Interchanged number: 58
  • Sum: 85 + 58 = 143

The solution is correct. Alternatively, if we had assumed y – x = 3, we would have found x = 5 and y = 8, resulting in the original number being 58.

Conclusion:

By systematically setting up and solving a system of linear equations, we determined that the original two-digit number could be either 85 or 58. Both numbers satisfy the conditions given in the problem. The problem highlights the importance of careful variable definition and the application of basic algebraic principles to solve real-world problems. This approach can be extended to similar problems involving multi-digit numbers or more complex relationships between the digits. The solution demonstrates the power of analytical thinking in resolving mathematical puzzles.

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