In a certain code, ‘GIVE’ is written as ‘5137’ and ‘BAT’ is written as ‘924’.…

Concept: In a letter-coding puzzle, every letter of the alphabet is assigned one fixed digit (or symbol). To decode an unfamiliar word, first read off the letter-to-digit correspondence from the coded words that are already given, then apply that same table, letter by letter, to the new word.

Application:

1. From 'GIVE' = '5137', match each letter to the digit in the same position: G → 5, I → 1, V → 3, E → 7.

2. From 'BAT' = '924', match each letter to the digit in the same position: B → 9, A → 2, T → 4.

3. The word to decode, 'GATE', is made up entirely of letters that already have a known code: G, A, T, E.

4. Substitute each letter of 'GATE' with its code, keeping the same left-to-right order: G → 5, A → 2, T → 4, E → 7.

5. Write the digits in that order to get the coded word: 5-2-4-7, that is, '5247'.

Cross-check: Decoding '5247' using the same G=5, A=2, T=4, E=7 correspondence gives back G-A-T-E, and no letter shares a digit with any other letter used across 'GIVE' and 'BAT', so the mapping has no contradiction.

Result: So in this code, 'GATE' is written as '5247'.