Finding the Required Two-Digit Number

Video Explanation

Question

A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from 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:

\[ 10x + y = 6(x + y) + 4 \]

\[ 10x + y = 6x + 6y + 4 \]

\[ 4x – 5y = 4 \quad (1) \]

Step 4: Form the Second Equation

If 18 is subtracted, 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 (2):

\[ x = y + 2 \]

Substitute in equation (1):

\[ 4(y + 2) – 5y = 4 \]

\[ 4y + 8 – 5y = 4 \]

\[ -y + 8 = 4 \]

\[ -y = -4 \]

\[ y = 4 \]

Step 6: Find the Value of x

\[ x = 4 + 2 \]

\[ x = 6 \]

Conclusion

Original number:

\[ 10x + y = 10(6) + 4 \]

\[ = 64 \]

\[ \boxed{64} \]

Final Answer (For Exam)

The required number is 64.