In the following number analogy, 243 is related to 578. Using the same pattern, 163 is related to which of the following numbers?

Introduction / Context:
This is a digit based number analogy where we must observe how individual digits of the first number are transformed to obtain the second number. The pair 243 and 578 encodes a pattern that uses simple addition applied separately to each digit. We are asked to apply the same transformation to the number 163 and then choose the correct resulting number from the given alternatives. Such questions test attention to detail in place value and digit wise operations.

Given Data / Assumptions:

• First number pair: 243 : 578
• Second first term: 163
• We need to find the second term in the same relation as 578 is to 243.
• We treat the numbers as three digit numbers with hundreds, tens, and units places.

Concept / Approach:
Instead of looking for a single formula connecting 243 to 578, it is often useful to compare corresponding digits. When we place 2, 4, 3 under 5, 7, 8 we notice the differences between each pair of digits. If the differences are consistent and simple, we can apply the same differences to the digits of 163. This is a common style of reasoning question where the transformation is applied place by place rather than to the whole number at once.

Step-by-Step Solution:
Step 1: Write the first pair in digit form. 243 has digits 2, 4, and 3. 578 has digits 5, 7, and 8. Step 2: Compare corresponding digits. From 2 to 5 the change is +3. From 4 to 7 the change is also +3. From 3 to 8 the change is +5. Step 3: Conclude the rule: add 3 to the first digit, add 3 to the second digit, and add 5 to the third digit of the number. Step 4: Apply this rule to the number 163. The first digit 1 becomes 1 + 3 = 4. The second digit 6 becomes 6 + 3 = 9. The third digit 3 becomes 3 + 5 = 8. Step 5: Combine the transformed digits to obtain the new number 498.

Verification / Alternative check:
To verify, apply the rule once more to the original pair. Starting from 243, adding 3 to the first digit gives 5, adding 3 to the second gives 7, and adding 5 to the third gives 8, which matches 578 exactly. The rule is therefore consistent with the given pair. For 163, we correctly obtained 498. Any different rule would either fail to produce 578 from 243 or would not give a clean mapping for 163. Thus 498 is the unique number that maintains the same digit wise transformation.

Why Other Options Are Wrong:
Numbers like 291, 326, and 347 do not result from adding plus three, plus three, and plus five to the digits of 163. For example, 291 would require changes of plus one, minus seven, and plus eight, which do not match the pattern seen in the first pair. A proper analogy demands that one single rule works for both pairs. Since those alternative options would force a different inconsistent rule, they are incorrect. Only 498 preserves exactly the same transformations that connect 243 and 578.

Common Pitfalls:
A typical mistake is to search for a whole number formula such as multiplying 243 by a constant or adding a large fixed number. This often leads to fractional or inconsistent results. Another pitfall is to change the order of digits through reversal or rotation, which is not supported by the first pair. For digit based analogies, always try the simplest possibility first: compare digits one by one and look for constant additions or subtractions. This method usually reveals the intended pattern quickly.

Final Answer:
Following the same digit wise pattern as 243 : 578, the number related to 163 is 498.