Finding the Two-Digit Number

Video Explanation

Question

The sum of the digits of a two-digit number is 13. If the number is subtracted from the one obtained by interchanging the digits, the result is 45. Find the number.

Solution

Step 1: Let the Variables

Let the tens digit = \(x\)

Let the units digit = \(y\)

Step 2: Form the Numbers

Original number = \(10x + y\)

Interchanged number = \(10y + x\)

Step 3: Form the Equations

Sum of digits is 13:

\[ x + y = 13 \quad (1) \]

Given condition:

\[ (10y + x) – (10x + y) = 45 \]

\[ 10y + x – 10x – y = 45 \]

\[ 9y – 9x = 45 \]

\[ y – x = 5 \quad (2) \]

Step 4: Solve the Equations

From equation (2):

\[ y = x + 5 \]

Substitute in equation (1):

\[ x + (x + 5) = 13 \]

\[ 2x + 5 = 13 \]

\[ 2x = 8 \]

\[ x = 4 \]

Step 5: Find the Value of y

\[ y = 4 + 5 \]

\[ y = 9 \]

Conclusion

Required number:

\[ 10x + y = 10(4) + 9 \]

\[ = 49 \]

\[ \boxed{49} \]

Final Answer (For Exam)

The required number is 49.