Question

The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2?

• A. 4
• B. 6
• C. 8
• D. 10
• E.

Hint:

In problems involving ratios and digit manipulation, start by expressing relationships between the digits algebraically and use the given conditions to form equations.

Answer 1

The Correct Option is C

Step 1: Define Variables Let the two-digit number be represented as \( 10x + y \), where: - \( x \) is the tens digit. - \( y \) is the units digit. Given that the ratio between the digits is \( 1:2 \), we write: \[ \frac{x}{y} = \frac{1}{2} \quad \Rightarrow \quad y = 2x. \] Step 2: Number Obtained by Interchanging the Digits When the digits are interchanged, the new number becomes: \[ 10y + x. \] Step 3: Establish the Difference Condition The difference between the original number and the interchanged number is given as 36: \[ (10x + y) - (10y + x) = 36. \] Step 4: Simplify the Equation \[ 10x + y - 10y - x = 36. \] \[ 9x - 9y = 36. \] Dividing throughout by 9: \[ x - y = 4. \] Step 5: Substitute \( y = 2x \) \[ x - 2x = 4. \] \[ -x = 4. \] \[ x = -4. \] Since digits cannot be negative, let's reconsider the absolute difference: \[ |(10x + y) - (10y + x)| = 36. \] \[ |9x - 9y| = 36. \] \[ |x - y| = 4. \] Step 6: Solve for \( x \) and \( y \) Using \( y = 2x \), substitute into \( |x - y| = 4 \): \[ |x - 2x| = 4. \] \[ |-x| = 4. \] \[ x = 4. \] Since \( x = 4 \), we find: \[ y = 2(4) = 8. \] Step 7: Compute the Sum and Difference of the Digits - Sum of the digits: \[ x + y = 4 + 8 = 12. \] - Difference of the digits: \[ y - x = 8 - 4 = 4. \] Step 8: Compute the Final Difference \[ (x + y) - (y - x) = 12 - 4 = 8. \] Final Answer: The difference between the sum and the difference of the digits is \( \boxed{8} \). .