Finding the Required Two-Digit Number

Video Explanation

Question

The sum of the digits of a two-digit number is 9. Nine times the number is twice the number obtained by reversing the digits. 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 Equations

Sum of digits:

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

Nine times the number equals twice the reversed number:

\[ 9(10x + y) = 2(10y + x) \]

\[ 90x + 9y = 20y + 2x \]

\[ 90x – 2x = 20y – 9y \]

\[ 88x = 11y \]

\[ 8x = y \quad (2) \]

Step 4: Solve the Equations

Substitute equation (2) into equation (1):

\[ x + 8x = 9 \]

\[ 9x = 9 \]

\[ x = 1 \]

Step 5: Find the Value of y

\[ y = 8(1) \]

\[ y = 8 \]

Conclusion

Original number:

\[ 10x + y = 10(1) + 8 \]

\[ = 18 \]

\[ \boxed{18} \]

Final Answer (For Exam)

The required number is 18.