The sum of the digits of a two-digit number is 10. When the digits are reversed, the number decreases by 54. What is the new (reversed) number?

Introduction:
This question involves a two-digit number whose digits satisfy certain conditions. It tests your ability to translate verbal statements about digits into algebraic equations and then solve for the unknown number.

Given Data / Assumptions:

• Original number is a two-digit number.
• Sum of its digits is 10.
• When the digits are reversed, the value of the number decreases by 54.
• We must find the reversed number (the changed number).

Concept / Approach:
Let the original number have tens digit a and units digit b. Then: Original number = 10a + b. Reversed number = 10b + a. We use the given conditions to derive equations:

• Digit sum: a + b = 10.
• Difference: (10a + b) - (10b + a) = 54.

Then we solve for a and b and obtain the reversed number.

Step-by-Step Solution:
Step 1: Use the difference condition. Original - Reversed = 54. So: (10a + b) - (10b + a) = 54. Simplify: 10a + b - 10b - a = 54 ⇒ 9a - 9b = 54. Divide by 9: a - b = 6. (Equation 1) Step 2: Use the digit sum condition. a + b = 10. (Equation 2) Step 3: Solve the system of equations. Add Equation 1 and Equation 2: (a - b) + (a + b) = 6 + 10 ⇒ 2a = 16 ⇒ a = 8. Substitute a = 8 into Equation 2: 8 + b = 10 ⇒ b = 2. Step 4: Form the numbers. Original number = 10a + b = 10 * 8 + 2 = 82.
Reversed number = 10b + a = 10 * 2 + 8 = 28.

Verification / Alternative Check:
Check the conditions: Digit sum of 82: 8 + 2 = 10 (correct). Reversing 82 gives 28. 82 - 28 = 54 (correct). So the reversed number is indeed 28.

Why Other Options Are Wrong:
46, 19, and 37 do not come from digits that sum to 10 while also satisfying the 54 difference condition. 82 is the original number, not the changed one. Only 28 satisfies both given conditions as the reversed number.

Common Pitfalls:
Students sometimes confuse which number is larger and may incorrectly set up the difference as reversed minus original. Others may forget that a is the tens digit and b is the units digit, leading to wrong algebraic expressions. Carefully defining digits and checking both conditions avoids these errors.

Final Answer:
The reversed (changed) number is 28.