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

Answer 1

The Correct Option is C

Define Variables: 

Let the two-digit number be \( 10x + y \), where:

• \( x \) is the tens digit.
• \( y \) is the units digit.

The ratio between the digits is \( 1:2 \), so:

\[ \frac{x}{y} = \frac{1}{2} \Rightarrow y = 2x \]

2. Number Obtained by Interchanging the Digits:

The number obtained by interchanging the digits is \( 10y + x \).

3. Difference Between the Two Numbers:

The difference between the original number and the interchanged number is 36:

\[ (10x + y) - (10y + x) = 36 \]

Simplify:

\[ 10x + y - 10y - x = 36 \]

Divide through by 9:

\[ 9x - 9y = 36 \]

\[ x - y = 4 \quad (1) \]

4. Substitute \( y = 2x \) into Equation (1):

\[ x - 2x = 4 \]

\[ -x = 4 \]

\[ x = -4 \]

This result is not valid because digits cannot be negative. Let’s re-examine the problem.

5. Re-evaluate the Difference:

The absolute difference between the two numbers is 36:

\[ |(10x + y) - (10y + x)| = 36 \]

6. Use the Ratio \( y = 2x \):

Substituting \( y = 2x \) into the equation:

\[ |9x - 9y| = 36 \]

\[ |x - y| = 4 \quad (2) \]

\[ |x - 2x| = 4 \]

\[ |-x| = 4 \]

\[ x = 4 \]

\[ y = 2x = 8 \]

7. Calculate the Sum and Difference of the Digits:

• Sum of the digits: \( x + y = 4 + 8 = 12 \)
• Difference of the digits: \( y - x = 8 - 4 = 4 \)

8. Find the Difference Between the Sum and the Difference:

\[ (x + y) - (y - x) = 12 - 4 = 8 \]

Final Answer: The difference between the sum and the difference of the digits is 8.