Finding the Required Two-Digit Number

Video Explanation

Question

A two-digit number is 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.

Solution

Step 1: Let the Variables

Let the tens digit = \(x\)

Let the units digit = \(y\)

Step 2: Form the Number

Original number = \(10x + y\)

Reversed number = \(10y + x\)

Step 3: Form the First Equation

Given:

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

\[ 10x + y = 4x + 4y \]

\[ 6x – 3y = 0 \]

\[ 2x – y = 0 \quad (1) \]

Step 4: Form the Second Equation

If 18 is added, digits are reversed:

\[ 10x + y + 18 = 10y + x \]

\[ 10x + y + 18 – 10y – x = 0 \]

\[ 9x – 9y + 18 = 0 \]

\[ x – y = -2 \quad (2) \]

Step 5: Solve the Equations

From equation (1):

\[ y = 2x \]

Substitute in equation (2):

\[ x – 2x = -2 \]

\[ -x = -2 \]

\[ x = 2 \]

Step 6: Find the Value of y

\[ y = 2(2) \]

\[ y = 4 \]

Conclusion

Original number:

\[ 10x + y = 10(2) + 4 \]

\[ = 24 \]

\[ \boxed{24} \]

Final Answer (For Exam)

The required number is 24.