Decoding the Letter Pattern: GRANT to TIZMG

The question asks us to find the coding pattern used to transform GRANT into TIZMG and apply it to code the word ROUGH.

Identifying the Coding Logic

Let's analyze the relationship between the letters of GRANT and TIZMG:

• G (7th letter) corresponds to T (20th letter)
• R (18th letter) corresponds to I (9th letter)
• A (1st letter) corresponds to Z (26th letter)
• N (14th letter) corresponds to M (13th letter)
• T (20th letter) corresponds to G (7th letter)

We observe that in each pair, the sum of the positional values of the letters equals 27. For example, for G, its position is 7, and T's position is 20. Calculation: $7 + 20 = 27$. This indicates the code uses the concept of opposite letters in the English alphabet (where A is opposite Z, B is opposite Y, etc.).

Applying the Pattern to ROUGH

To find the code for ROUGH, we will find the opposite letter for each letter in ROUGH:

• R: The position of R is 18. The opposite letter's position is calculated as $27 - 18 = 9$. The 9th letter is I.
• O: The position of O is 15. The opposite letter's position is calculated as $27 - 15 = 12$. The 12th letter is L.
• U: The position of U is 21. The opposite letter's position is calculated as $27 - 21 = 6$. The 6th letter is F.
• G: The position of G is 7. The opposite letter's position is calculated as $27 - 7 = 20$. The 20th letter is T.
• H: The position of H is 8. The opposite letter's position is calculated as $27 - 8 = 19$. The 19th letter is S.

Combining these opposite letters, we get the code for ROUGH as ILFTS.

Final Coded Word

Therefore, ROUGH is coded as ILFTS.