Finding the Two-Digit Number

Video Explanation

Question

The sum of digits of a two-digit number is 15. The number obtained by reversing the digits exceeds the given number by 9. Find the given number.

Solution

Step 1: Let the Variables

Let the tens digit = \(x\)

Let the units digit = \(y\)

Step 2: Form the Numbers

Original number = \(10x + y\)

Reversed number = \(10y + x\)

Step 3: Form the Equations

Sum of digits:

\[ x + y = 15 \quad (1) \]

Reversed number exceeds original by 9:

\[ (10y + x) – (10x + y) = 9 \]

\[ 10y + x – 10x – y = 9 \]

\[ 9y – 9x = 9 \]

\[ y – x = 1 \quad (2) \]

Step 4: Solve the Equations

From equation (2):

\[ y = x + 1 \]

Substitute in equation (1):

\[ x + (x + 1) = 15 \]

\[ 2x + 1 = 15 \]

\[ 2x = 14 \]

\[ x = 7 \]

Step 5: Find the Value of y

\[ y = 7 + 1 \]

\[ y = 8 \]

Conclusion

Original number:

\[ 10x + y = 10(7) + 8 \]

\[ = 78 \]

\[ \boxed{78} \]

Final Answer (For Exam)

The required number is 78.