In a certain code language CHARCOAL is coded as 45164913 and MORALE is coded as 296137. Using the same letter–digit substitution, how will the word REAL be coded in that language?

Introduction / Context:
This question involves a consistent mapping of letters to digits. We are told that CHARCOAL corresponds to 45164913 and MORALE to 296137. Our task is to deduce the digit associated with each letter used in these words and then apply the mapping to the word REAL to produce its code. This is a direct substitution problem, common in coding–decoding sections.

Given Data / Assumptions:

• CHARCOAL is written as 45164913.
• MORALE is written as 296137.
• Each letter is always coded by the same single digit.
• Letter order is preserved when writing the code.
• The word REAL uses only letters already seen in the examples.

Concept / Approach:
We align each word with its numeric code to create letter–digit pairs. By observing repeated letters (like C and A in CHARCOAL) we confirm consistency of the mapping. Next, we read off the digits for the letters R, E, A and L from this mapping and combine them in order to get the code for REAL.

Step-by-Step Solution:
Step 1: Align CHARCOAL and its code: C H A R C O A L corresponds to 4 5 1 6 4 9 1 3. Step 2: Extract mappings: C -> 4, H -> 5, A -> 1, R -> 6, O -> 9, L -> 3. Step 3: Notice that C appears twice and is coded as 4 both times; A appears twice and is coded as 1 both times, confirming consistency. Step 4: Align MORALE and its code: M O R A L E corresponds to 2 9 6 1 3 7. Step 5: From MORALE, we reinforce mappings already known: O -> 9, R -> 6, A -> 1, L -> 3, and also get a new mapping: M -> 2 and E -> 7. Step 6: Now we must code REAL, whose letters are R, E, A and L. Step 7: Substitute using the mapping: R -> 6, E -> 7, A -> 1, L -> 3, giving 6 7 1 3. Step 8: Therefore, REAL is coded as 6713.

Verification / Alternative check:
Check once more: R (6) from both CHARCOAL and MORALE, E (7) from MORALE, A (1) from both words, and L (3) from both words. The sequence 6713 uses the correct digits in the correct letter order. No other option matches this exact mapping without contradicting the established substitutions.

Why Other Options Are Wrong:
Option A: 8519 uses digits not matching the mapping for R, E, A and L. Option C: 6513 miscodes E and A, swapping their digits relative to the correct order. Option D: 6719 replaces the last digit (for L) with 9, which belongs to O instead. Option E: 8613 changes the digit for E incorrectly.

Common Pitfalls:
Some students try to deduce a mathematical pattern from the digits instead of recognising that they are mere labels for letters. Others misalign letters when building the mapping table, causing one or two incorrect pairings that lead to the wrong option. Writing a clear mapping list and checking repeated letters prevents such errors.

Final Answer:
The correct code for REAL in this language is 6713.