Question

Sum of the digits of a two-digit number is 12. The number formed by interchanging the digits is 18 more than original number. Find the number.

Hint:

Always verify your answer with the conditions in the problem. The number is 57. Sum of digits: 5+7=12. Interchanged number is 75. Difference: 75-57=18. Both conditions are satisfied.

Answer 1

Step 1: Understanding the Concept:
This is a word problem that can be solved by setting up a system of two linear equations in two variables representing the digits of the number.
Step 2: Key Formula or Approach:
Let the tens digit of the two-digit number be \(x\) and the units digit be \(y\).
The original number can be expressed as \(10x + y\).
The number formed by interchanging the digits is \(10y + x\).
Step 3: Detailed Explanation:
From the first condition, the sum of the digits is 12.
\[ x + y = 12 \quad \ldots (1) \] From the second condition, the new number is 18 more than the original number.
\[ (10y + x) = (10x + y) + 18 \] Rearrange the terms to form a second linear equation:
\[ 10y - y + x - 10x = 18 \] \[ 9y - 9x = 18 \] Divide the entire equation by 9:
\[ y - x = 2 \quad \ldots (2) \] Now we have a system of two linear equations:
1) \( x + y = 12 \)
2) \( -x + y = 2 \)
Adding equation (1) and equation (2):
\[ (x + y) + (-x + y) = 12 + 2 \] \[ 2y = 14 \] \[ y = 7 \] Substitute the value of \(y = 7\) into equation (1):
\[ x + 7 = 12 \] \[ x = 12 - 7 \] \[ x = 5 \] The tens digit is 5 and the units digit is 7.
Step 4: Final Answer:
The original number is \(10x + y = 10(5) + 7 = 57\).