In a certain code language, the word EXPLAIN is written as BUMIXFK by shifting each letter three places backward in the alphabet. Using the same rule, how will the word GYM be coded?

Introduction / Context:
This is another Caesar cipher style coding question where each letter is replaced by a letter that is a fixed number of positions away in the alphabet. The word EXPLAIN becomes BUMIXFK when each letter is shifted three places backward. We must apply this same uniform shift to the word GYM to find its coded form.

Given Data / Assumptions:

• EXPLAIN is coded as BUMIXFK.
• The alphabet positions A = 1 through Z = 26 are considered.
• The code is formed by shifting letters backward by three positions.
• We need to code the word GYM using the same rule.
• Wrap around is used if the shift passes A.

Concept / Approach:
We first confirm the shift using EXPLAIN and BUMIXFK. Each letter of EXPLAIN maps to a letter three positions earlier. Once this is confirmed, we apply the same backward shift of three positions to each letter in GYM, and then combine the results to get the final coded word.

Step-by-Step Solution:
Step 1: Check the shift for EXPLAIN. E (5) to B (2) is a shift of −3. X (24) to U (21) is −3. P (16) to M (13) is −3. L (12) to I (9) is −3. A (1) to X (24) is also −3 with wrap around. I (9) to F (6) is −3. N (14) to K (11) is −3.Step 2: Since each letter is shifted three positions backward, the rule is confirmed.Step 3: Now apply this rule to GYM. Find the positions: G = 7, Y = 25, M = 13.Step 4: Shift each letter three places backward. G (7) minus 3 gives D (4). Y (25) minus 3 gives V (22). M (13) minus 3 gives J (10).Step 5: Combine these letters to form the coded word DVJ.

Verification / Alternative check:
To verify, shift the letters of DVJ three positions forward: D → G, V → Y and J → M, which reconstructs GYM exactly. This backward and forward check confirms that the shift has been applied correctly to each letter.

Why Other Options Are Wrong:
The options YHN, UJM, IKL and XJL do not result from shifting G, Y and M three positions backward. Some may involve different shifts or mixed directions, but they do not correspond to a consistent −3 shift from each letter. Therefore, only DVJ adheres to the established coding rule derived from EXPLAIN.

Common Pitfalls:
Students often miscount three places backward or accidentally shift forward instead. Another common error is to apply different shifts to different letters. Carefully computing numeric positions and subtracting three from each one avoids these mistakes.

Final Answer:
Using the same three letter backward shift as in EXPLAIN, the word GYM is coded as DVJ.