The numbers 382, 473, 568, 728, 847 and 629 are written so that the digits of each number are arranged in ascending order. Which pair of these numbers then has the smallest difference between their new forms?

Introduction / Context:
This question tests your ability to work with place values and digit rearrangement in three digit numbers. When digits are rearranged in a specific way, the new numbers can be compared to find differences. You are asked to follow a clear rule for re writing each number and then determine which pair exhibits the smallest difference between their transformed values.

Given Data / Assumptions:
- Original numbers: 382, 473, 568, 728, 847, and 629. - Transformation rule: write each number so that its digits are in ascending order. - Examples given: 382 becomes 238, 473 becomes 347. - We must compare the transformed values of the pairs listed in the options.

Concept / Approach:
The approach is to apply the same digit sorting rule to each number, then calculate the absolute difference between the transformed numbers in each given pair. The digits of a three digit number can be viewed as hundreds, tens, and units. Sorting them in ascending order changes the place value contribution of each digit, which changes the value of the number. Once all new forms are known, a simple subtraction identifies the pair with the smallest difference.

Step-by-Step Solution:
Step 1: Transform 382. Digits are 3, 8, and 2. In ascending order they are 2, 3, 8, giving 238. Step 2: Transform 473. Digits are 4, 7, and 3. In ascending order they are 3, 4, 7, giving 347. Step 3: Transform 568. Digits 5, 6, and 8 are already in ascending order, so the new form is 568. Step 4: Transform 728. Digits 7, 2, and 8 in ascending order are 2, 7, 8, giving 278. Step 5: Transform 847. Digits 8, 4, and 7 in ascending order are 4, 7, 8, giving 478. Step 6: Transform 629. Digits 6, 2, and 9 in ascending order are 2, 6, 9, giving 269. Step 7: Compute differences for the pairs in the options: - For 473 and 382: 347 minus 238 equals 109. - For 629 and 728: 278 minus 269 equals 9. - For 629 and 847: 478 minus 269 equals 209. - For 728 and 847: 478 minus 278 equals 200. Step 8: The smallest difference among these is 9, corresponding to the pair 629 and 728.

Verification / Alternative check:
As a quick mental check, focus only on the transformed numbers, not the originals. The transformed list is 238, 347, 568, 278, 478, and 269. If you sort these, the closest pair numerically appears around 269 and 278. These are the transformed forms of 629 and 728. Confirming this pair yields a difference of 9, and no other pair produces a smaller difference. This confirms that the chosen option is consistent and unique.

Why Other Options Are Wrong:
473 and 382: Their new values differ by 109, which is much larger than 9. 629 and 847: The difference between 269 and 478 is 209, one of the largest differences. 728 and 847: Their transformed values 278 and 478 differ by 200, again far from the minimum.

Common Pitfalls:
Learners sometimes compare the original numbers instead of the transformed versions, which completely changes the result. Another common mistake is incorrect digit sorting, such as treating 473 as 437 rather than 347. Finally, some candidates forget to take absolute differences and may consider negative values or reversed subtraction, but since we care about magnitude, the order of subtraction does not matter as long as the absolute difference is used.

Final Answer:
The pair of numbers whose transformed forms have the smallest difference is 629 and 728.