Two-Digit Number and Its Reverse

Video Explanation

Question

The sum of a two-digit number and the number obtained by reversing its digits is 99. If the digits differ by 3, 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 = \(10x + y\)

Reversed number = \(10y + x\)

Step 3: Form the First Equation

\[ (10x + y) + (10y + x) = 99 \]

\[ 11x + 11y = 99 \]

\[ x + y = 9 \quad (1) \]

Step 4: Digits Differ by 3

Two possible cases:

Case 1: \[ x – y = 3 \quad (2) \]

Case 2: \[ y – x = 3 \quad (3) \]

Case 1: Solve (1) and (2)

x + y = 9

x – y = 3

Add equations:

\[ 2x = 12 \]

\[ x = 6 \]

\[ y = 3 \]

Number = \(63\) —

Case 2: Solve (1) and (3)

x + y = 9

y – x = 3

Add equations:

\[ 2y = 12 \]

\[ y = 6 \]

\[ x = 3 \]

Number = \(36\)

Conclusion

The required numbers are:

\[ \boxed{63 \text{ and } 36} \]

Final Answer (For Exam)

The numbers are 63 and 36.