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. Find both the numbers.
The two numbers are 85 and 58.
Explanation
By representing the two-digit number algebraically as 10x + y, we can translate the word problem into a system of linear equations. The condition that the digits differ by 3 gives |x - y| = 3, and the sum condition upon interchanging gives x + y = 13. Solving these equations yields the digits, and thus the numbers.
Solution Steps
Step 1: Let the tens digit of the original number be x and the units digit be y. The original number is 10x + y.
Step 2: When the digits are interchanged, the new number is 10y + x. The sum of both numbers is 143, so (10x + y) + (10y + x) = 143, which simplifies to 11x + 11y = 143, or x + y = 13.
Step 3: The digits differ by 3, which gives two possibilities: x - y = 3 or y - x = 3.
Step 4: Case 1: If x - y = 3, adding this to x + y = 13 gives 2x = 16, so x = 8. Substituting x in x + y = 13 gives y = 5. The original number is 85, and the interchanged number is 58.
Step 5: Case 2: If y - x = 3, adding this to x + y = 13 gives 2y = 16, so y = 8. Substituting y in x + y = 13 gives x = 5. The original number is 58, and the interchanged number is 85.
Step 6: In both cases, the two numbers are 85 and 58.