In this number analogy, 850 becomes 863 by a simple operation on its digits. Using the same rule, what number should replace the question mark in 430 : ?

Introduction / Context:
This question is a three digit number analogy where the second number is obtained from the first by a predictable operation involving its digits. The pair 850 : 863 shows that 863 is formed from 850 in some consistent way. Your job is to identify this rule and then apply it to 430 in order to find the correct related number from the options. Such problems often use digit sums or simple arithmetic adjustments to generate the second number.

Given Data / Assumptions:

The known pair is 850 and 863.
The unknown pair is 430 and ?.
We work with the digits of the three digit numbers individually.
Options for the missing number are 437, 442, 435 and 451.
We assume the transformation uses basic digit operations suitable for mental calculation in an exam setting.

Concept / Approach:
A common trick is to look at the difference between the numbers. From 850 to 863, the increase is 13. Next, consider the sum of the digits of 850: 8 + 5 + 0 equals 13. This suggests the rule: second number = first number + sum of its digits. If this holds for 850, we should apply the same rule to 430. The sum of the digits of 430 is then added to 430, and we check which of the options matches the result.

Step-by-Step Solution:
Step 1: Compute the digit sum of 850. Add 8 + 5 + 0 to get 13. Step 2: Add this sum to the original number: 850 + 13 = 863. This matches the second number, confirming the rule. Step 3: Now apply the same procedure to 430. First, find the digit sum: 4 + 3 + 0 = 7. Step 4: Add this digit sum to 430: 430 + 7 = 437. Step 5: Compare this result with the options and see that 437 is present. Step 6: Conclude that 430 must be related to 437 in the same way that 850 is related to 863.

Verification / Alternative check:
To verify the robustness of the rule, imagine applying it to another test number. For example, if we used 512, the sum of its digits is 5 + 1 + 2 = 8, and 512 + 8 = 520. This behaviour is simple and symmetric. Re checking our original example confirms that 8 + 5 + 0 equals 13 and 850 + 13 equals 863, so the pattern is correct. For 430, 4 + 3 + 0 equals 7, and 430 + 7 equals 437, which matches exactly one of the answer choices.

Why Other Options Are Wrong:
Option 442 would require adding 12 to 430, but the sum of the digits of 430 is 7, not 12. Option 435 would require adding 5, which is again not the digit sum. Option 451 would require adding 21, which does not match any simple digit operation. Only 437 is produced by the rule “original number plus sum of its digits” and therefore preserves the analogy correctly.

Common Pitfalls:
A common mistake is to look for a fixed difference and ignore the internal digit structure, or to miscalculate the digit sum. Another pitfall is to assume that the change must affect only one digit position, such as the units place, while the given pair clearly changes both tens and units. To avoid such errors, always compute and compare both differences and digit sums early in your analysis. If the difference matches the sum of digits, that is usually a strong candidate for the underlying rule in number analogies like this one.

Final Answer:
Since 863 is obtained from 850 by adding the digit sum 13, and 437 is obtained from 430 by adding its digit sum 7, the correct related number is 437.